
\( |\partial E|_\mathbb {G}(\mathcal {U} (p,r))>0, \)  for all \(r>0\)

there exists \(\lim _{r\rightarrow 0} {\mathchoice{\hspace{4.25pt}\hrule height.4ptwidth 6ptdepth0pt}{\hspace{0.0pt}}{-}{9}.7pt} {\hspace{4.25pt}\hrule height.4pt width 4.3ptdepth0pt}\hspace{-7.0pt}\)    U (p,r) E   d |E|G\(\item \) | r0   
   U (p,r) E   d |E|G| =1\(\)
The reduce boundary of a set \(E\subset \mathbb {G}\)  is invariant under group translations, i.e.
\(q\in \partial ^*_\mathbb {G}E \qquad \mbox{if and only if}\qquad \tau _pq\in \partial ^*_\mathbb {G}(\tau _pE)\) 
and also \(\nu _E (q)=\nu _{\tau _pE} (\tau _p q)\) .
Lemma 2.3 (Differentiation Lemma, [1]})  If \(E\subset \mathbb {G}\)  is a \(\mathbb {G}\) -Caccioppoli set, then

\(\lim _{r\rightarrow 0} {\mathchoice{\hspace{4.25pt}\hrule height.4ptwidth 6ptdepth0pt}{\hspace{0.0pt}}{-}{9}.7pt} {\hspace{4.25pt}\hrule height.4pt width 4.3ptdepth0pt}\hspace{-7.0pt}\) 
   U (p,r) E   d |E|G=E (p),    for  |E|G-a.e.  p

hence \(|\partial E|_\mathbb {G}\)  is concentrated on the reduced boundary \(\partial ^*_\mathbb {G}E\) .
The perimeter measure equals a constant times the spherical (\(\mathfrak {q}-1\) )-dimensional Hausdorff measure restricted to the reduced boundary, indeed
Theorem 2.4 ([2]}, Theorem 4.18) 
Let \(\mathbb {G}\)  be a Carnot group of step 2, endowed with the invariant distance \(d\) . If \(E \subset \mathbb {G}\)  is a \(\mathbb {G}\) -Caccioppoli set, then
\(|\partial E|_\mathbb {G}= c \, \mathcal {S}^{\mathfrak {q}-1} \, \mathop {\hbox{\vrule height 7pt width .5pt depth 0pt\vrule height .5pt width 6pt depth 0pt}}\nolimits \, \partial ^*_\mathbb {G}E\) 
where \(\mathfrak {q}\)  is homogeneous dimension of \(\mathbb {G}\)  defined in Section \(\ref {Carnotinizio}\) .

At each point of the reduced boundary of a \(\mathbb {G}\) -Caccioppoli set there is a (generalized) tangent group:
Theorem 2.5 (Blow-up Theorem, [3]})  Let \(\mathbb {G}\)  be a Carnot group of step 2 and let \(E\subset \mathbb {G}\)  be a set with locally finite \(\mathbb {G}\) -perimeter. If \(p\in \partial ^*_\mathbb {G}E\)  then
\(\lim _{r\rightarrow 0 } \chi _{ E_{r,p} } = \chi _{ S^+_\mathbb {G}(\nu _E(p)) } \quad \mbox{in } \mathcal {L}^1_{loc} (\mathbb {G})\) 
where \(E_{r,p}:= \delta _{1/r} (\tau _{p^{-1}} E) =\lbrace  q\, :\, \tau _p(\delta _r (q)) \in E\rbrace \)  and
\(S^+_\mathbb {G}(\nu _E(p)) :=\lbrace  q=(q^1,q^2) \in \mathbb {G}\, |\, \langle \nu _E (p), q^1 \rangle \ge 0 \rbrace .\) 
Moreover for all \(\delta >0\)
\(\lim _{r\rightarrow 0 } |\partial E_{r,p} |_\mathbb {G}(\mathcal {U}(0,\delta ))=|\partial S^+_\mathbb {G}(\nu _E (p))|_\mathbb {G}(\mathcal {U}(0,\delta ))\) 
and
\(|\partial S^+_\mathbb {G}(\nu _E (p))|_\mathbb {G}(\mathcal {U}(0,\delta )) = \mathcal {H}^{N-1} (T_\mathbb {G}^g (\nu _E(0))\cap \mathcal {U}(0,\delta ))\) 
where \(T_\mathbb {G}^g (\nu _E(0)):= \lbrace  q=(q^1,q^2) \in \mathbb {G}\, |\, \langle \nu _E (0), Q^1 \rangle = 0 \rbrace \)  is the topological boundary of \(S^+_\mathbb {G}(\nu _E (0))\) .

Finally, as it is usual in the literature, we can also define the measure theoretic boundary \(\partial _{*,\mathbb {G}} E\) :
Definition 2.4 Let \(E\subset \mathbb {G}\)  be a measurable set. We say that \(p\)  belongs to measure theoretic boundary \(\partial _{*,\mathbb {G}} E\)  of \(E\)  if
\(\limsup _{r\rightarrow 0 ^+} \frac{\mathcal {L}^N (E\cap \mathcal {U} (p,r) ) }{ \mathcal {L}^{N} ( \mathcal {U} (p,r) )} >0 \qquad \mbox{and}\qquad \limsup _{r\rightarrow 0 ^+} \frac{\mathcal {L}^{N} (E^c\cap \mathcal {U}(p,r) ) }{ \mathcal {L}^{N} ( \mathcal {U}(p,r) )} >0.\) 

If \(E\subset \mathbb {G}\)  is \(\mathbb {G}\) -Caccioppoli set, then
\(\partial ^*_{\mathbb {G}} E \subset \partial _{*,\mathbb {G}} E \subset \partial E.\) 
Moreover, \(\mathcal {S}^{\mathfrak {q}-1} ( \partial _{*,\mathbb {G}} E - \partial ^*_{\mathbb {G}} E)=0.\)
Complementary subgroups and graphs
Definition 2.5 We say that \(\mathbb {W}\)  and \(\mathbb {M}\)  are complementary subgroups in \(\mathbb {G}\) if
\(\mathbb {W}\)  and \(\mathbb {M}\)  are homogeneous subgroups of \(\mathbb {G}\)  such that \(\mathbb {W}\cap \mathbb {M}= \lbrace  0 \rbrace \)  and
\(\mathbb {G}=\mathbb {W}\cdot \mathbb {M}.\) 
By this we mean that for every \(p\in \mathbb {G}\)  there are \(p_\mathbb {W}\in \mathbb {W}\)  and \(p_\mathbb {M}\in \mathbb {M}\)  such that \(p=p_\mathbb {W}p_\mathbb {M}\) .

The elements \(p_\mathbb {W}\in \mathbb {W}\)  and \(p_\mathbb {M}\in \mathbb {M}\)  such that \(p=p_\mathbb {W}\cdot p_\mathbb {M}\)  are unique because of \(\mathbb {W}\cap \mathbb {M}= \lbrace  0 \rbrace \)  and are denoted components of \(p\)  along \(\mathbb {W}\)  and \(\mathbb {M}\)  or projections of \(p\)  on \(\mathbb {W}\)  and \(\mathbb {M}\) .
The projection maps \(\mathbf {P}_\mathbb {W}:\mathbb {G}\rightarrow \mathbb {W}\)  and \(\mathbf {P}_\mathbb {M}:\mathbb {G}\rightarrow \mathbb {M}\)  defined
\(\mathbf {P}_\mathbb {W}(p)=p_\mathbb {W}, \qquad \mathbf {P}_\mathbb {M}(p)=p_\mathbb {M}, \qquad \text{for all $p\in \mathbb {G}$}\) 
are polynomial functions (see Proposition 2.2.14 in [4]}) if we identify \(\mathbb {G}\)  with \(\mathbb {R}^N\) , hence are \(\infty \) . Nevertheless in general they are not Lipschitz maps, when \(\mathbb {W}\)  and \(\mathbb {M}\)  are endowed with the restriction of the left invariant distance \(d\)  of \(\mathbb {G}\)  (see Example 2.2.15 in [4]}).
Remark 2.6 
The stratification of \(\mathbb {G}\)  induces a stratifications on the complementary subgroups \(\mathbb {W}\)  and \(\mathbb {M}\) . If \(\mathbb {G}=\mathbb {G}^1\oplus \dots \oplus \mathbb {G}^\kappa \)  then also \(\mathbb {W}= \mathbb {W}^1\oplus \dots \oplus \mathbb {W}^\kappa \) , \(\mathbb {M}=\mathbb {M}^1\oplus \dots \oplus \mathbb {M}^\kappa \)  and \(\mathbb {G}^i =\mathbb {W}^i \oplus \mathbb {M}^i\) . A subgroup is horizontal if it is contained in the first layer \(\mathbb {G}^1\) . If \(\mathbb {M}\)  is horizontal then the complementary subgroup \(\mathbb {W}\)  is normal.

Proposition 2.7 (see [6]}, Proposition 3.2) 
If \(\mathbb {W}\)  and \(\mathbb {M}\)  are complementary subgroups in \(\mathbb {G}\)  there is \(c_0=c_0(\mathbb {W}, \mathbb {M})\in (0,1)\)  such that for each \(p_\mathbb {W}\in \mathbb {W}\)  and \(p_\mathbb {M}\in \mathbb {M}\)
\(c_0(\Vert  p_\mathbb {W}\Vert +\Vert p_\mathbb {M}\Vert )\le \Vert  p_\mathbb {W}p_\mathbb {M}\Vert  \le \Vert  p_\mathbb {W}\Vert +\Vert p_\mathbb {M}\Vert \) 

Definition 2.6 We say that \(S\subset \mathbb {G}\)  is a left intrinsic graph or more simply a intrinsic graph if there are complementary subgroups \(\mathbb {W}\)  and \(\mathbb {M}\)  in \(\mathbb {G}\)  and \(\phi : \mathcal {O} \subset \mathbb {W}\rightarrow \mathbb {M}\)  such that
\(S=\mathrm {graph}\,(\phi ) :=\lbrace  a \phi (a):\, a\in \mathcal {O} \rbrace .\) 

Observe that, by uniqueness of the components along \(\mathbb {W}\)  and \(\mathbb {M}\) , if \(S=\mathrm {graph}\,(\phi )\)  then \(\phi \)  is uniquely determined among all functions from \(\mathbb {W}\)  to \(\mathbb {M}\) .
We call graph map of \(\phi \) , the function \(\Phi :\mathcal {O} \rightarrow \mathbb {G}\)  defined as
\(\Phi (a):= a \cdot \phi (a) \quad \mbox{for all } a\in \mathcal {O}.\) 
Hence \(S=\Phi (\mathcal {O} )\)  is equivalent to \(S=\mathrm {graph}\,(\phi )\) .
The concept of intrinsic graph is preserved by translation and dilation, i.e.
Proposition 2.8 (see Proposition 2.2.18 in [4]}) 
If \(S\)  is a intrinsic graph then, for all \(\lambda >0\)  and for all \(q\in \mathbb {G}\) , \(q \cdot S\)  and \(\delta _\lambda S\)  are intrinsic graphs. In particular, if \(S=\mathrm {graph}\,(\phi )\)  with \(\phi :\mathcal {O} \subset \mathbb {W}\rightarrow \mathbb {M}\) , then

For all \(\lambda >0\) ,
\(\delta _\lambda \left(\mathrm {graph}\,(\phi )\right) =\mathrm {graph}\,(\phi _\lambda )\) 
where
\(\phi _\lambda :\delta _\lambda \mathcal {O} \subset \mathbb {W}\rightarrow \mathbb {M}\)  and
\( \phi _\lambda (a):= \delta _\lambda \phi (\delta _{1/\lambda }a)\) , for \(a \in \delta _\lambda \mathcal {O}\) .

For any \(q\in \mathbb {G}\) ,
\(q \cdot \mathrm {graph}\,(\phi ) = \mathrm {graph}\,(\phi _q )\) 
where
\(\phi _q : \mathcal {O} _q \subset \mathbb {W}\rightarrow \mathbb {M}\)  is defined as
\(\phi _q (a):= (\mathbf {P}_\mathbb {M}(q^{-1}a))^{-1} \phi ( \mathbf {P}_\mathbb {W}(q^{-1}a))\) , for all \(a \in \mathcal {O}_q:=\lbrace  a\, :\, \mathbf {P}_\mathbb {W}(q^{-1}a)\in \mathcal {O} \rbrace \) .



Intrinsic differentiability
Definition 2.7 Let \(\mathbb {W}\)  and \(\mathbb {M}\)  be complementary subgroups in \(\mathbb {G}\) . Then \(\ell :\mathbb {W}\rightarrow \mathbb {M}\)  is intrinsic linear if \(\ell \)  is defined on all of \(\mathbb {W}\)  and if \(\mathrm {graph}\,(\ell ) \)  is a homogeneous subgroup of \(\mathbb {G}\) .

We use intrinsic linear functions to define intrinsic differentiability as in the usual definition of differentiability.
Definition 2.8 
Let \(\mathbb {W}\)  and \(\mathbb {M}\)  be complementary subgroups in \(\mathbb {G}\)  and let \(\phi :\mathcal {O} \subset \mathbb {W}\rightarrow \mathbb {M}\)  with \(\mathcal {O}\)  open in \(\mathbb {W}\) . For \(a\in \mathcal {O}\) , let \(p:=a\cdot \phi (a)\)  and \(\phi _{p^{-1}}: \mathcal {O} _{p^{-1}} \subset \mathbb {W}\rightarrow \mathbb {M}\)  be the shifted function defined in Proposition \(\ref {P2.2.18}\) .

We say that \(\phi \)  is intrinsic differentiable in \(a\) if the shifted function \(\phi _{p^{-1}}\)  is intrinsic differentiable in 0, i.e. if there is a intrinsic linear \(d\phi _a:\mathbb {W}\rightarrow \mathbb {M}\)  such that
\(\lim _{r\rightarrow 0^+}\sup _{0<\Vert b\Vert <r}\frac{\Vert  d\phi _{a} (b)^{-1} \phi _{p^{-1}} (b) \Vert }{\Vert b\Vert } =0.\) 
The function \(d\phi _a\)  is the intrinsic differential of \(\phi \)  at \(a\).

We say that \(\phi \)  is uniformly intrinsic differentiable in \(a_0\in \mathcal {O}\) or \(\phi \)  is u.i.d. in \(a_0\) if there exist a intrinsic linear function \(d\phi _{a_0}: \mathbb {W}\rightarrow \mathbb {M}\)  such that
\(\lim _{r\rightarrow 0^+}\sup _{\Vert a_0^{-1}a\Vert <r}\sup _{0<\Vert b\Vert <r}\frac{\Vert  d\phi _{a_0} (b)^{-1} \phi _{p^{-1}} (b) \Vert }{\Vert b\Vert } =0.\) 
Analogously, \(\phi \)  is u.i.d. in \(\mathcal {O}\)  if it is u.i.d. in every point of \(\mathcal {O}\) .


Remark 2.9 Definition REF  is a natural one because of the following observations.
(i) If \(\phi \)  is intrinsic differentiable in \(a\in \mathcal {O}\) , there is a unique intrinsic linear function \(d\phi _a\)  satisfying \((\ref {3.0})\) . Moreover \(\phi \)  is continuous at \(a\) . (See Theorem 3.2.8 and Proposition 3.2.3 in [8]}).
(ii) The notion of intrinsic differentiability is invariant under group translations. Precisely, let \(p:=a\phi (a), q:=b\phi (b)\) , then \(\phi \)  is intrinsic differentiable in \(a\)  if and only if \(\phi _{qp^{-1}} := (\phi _{p^{-1}})_{q}\)  is intrinsic differentiable in \(b\) .
(iii) It is clear, taking \(a=a_0\)  in (REF ), that if \(\phi \)  is uniformly intrinsic differentiable in \(a_0\)  then it is intrinsic differentiable in \(a_0\)  and \(d\phi _{a_0}\)  is the intrinsic differential of \(\phi \)  at \(a_0\).

The analytic definition of intrinsic differentiability of Definition \(\ref {d3.2.1}\)  has an equivalent geometric formulation. Indeed intrinsic differentiability in one point is equivalent to the existence of a tangent subgroup to the graph, i.e.
Theorem 2.10 (Theorem 3.2.8. in [8]}) 
Let \(\mathbb {W}, \mathbb {M}\)  be complementary subgroups in \(\mathbb {G}\)  and let \(\phi :\mathcal {O} \rightarrow \mathbb {M}\)  with \(\mathcal {O} \)  relatively open in \(\mathbb {W}\) . If \(\phi \)  is intrinsic differentiable in \(a\in \mathcal {O} \) , set \(\mathbb {T} :=\mathrm {graph}\,(d\phi _a)\) . Then

\(\mathbb {T}\)  is an homogeneous subgroup of \(\mathbb {G}\) ;

\(\mathbb {T}\)  and \(\mathbb {M}\)  are complementary subgroups in \(\mathbb {G}\) ;

\( p\cdot \mathbb {T}\)  is the tangent coset to \(\mathrm {graph}\,(\phi )\)  in \(p:=a\phi (a)\) .

Conversely, if \(p:=a\phi (a)\in \mathrm {graph}\,(\phi )\)  and if there is \(\mathbb {T}\)  such that \((1)\) , \((2)\) , \((3)\)  hold, then \(\phi \)  is intrinsic differentiable in \(a\)  and the differential \(d\phi _a : \mathbb {W}\rightarrow \mathbb {M}\)  is the unique intrinsic linear function such that \(\mathbb {T} :=\mathrm {graph}\,(d\phi _a)\) .

From now on we restrict our setting studying the notions of intrinsic differentiability and of uniform intrinsic differentiability for functions \(\phi : \mathbb {W}\rightarrow \mathbb {H}\)  when \(\mathbb {H}\)  is a horizontal subgroup. When \(\mathbb {H}\)  is horizontal, \(\mathbb {W}\)  is always a normal subgroup since, as observed in Remark \(\ref {rem2.2.1}\) , it contains the whole strata \(\mathbb {G}^2, \dots , \mathbb {G}^\kappa \) .
In this case, the more explicit form of the shifted function \(\phi _{P^{-1}} \)  allows a more explicit form of equations (REF ) and (REF ).
Proposition 2.11 (Theorem 3.5. in [10]}) 
Let \(\mathbb {H}\)  and \(\mathbb {W}\)  be complementary subgroups of \(\mathbb {G}\) , \(\mathcal {O}\)  open in \(\mathbb {W}\)  and \(\mathbb {H}\)  horizontal. Then \(\phi :\mathcal {O} \subset \mathbb {W}\rightarrow \mathbb {H}\)  is intrinsic differentiable in \(a_0\in \mathcal {O}\)  if and only if there is a intrinsic linear \(d\phi _{a_0}:\mathbb {W}\rightarrow \mathbb {H}\)  such that
\(\lim _{r\rightarrow 0^+}\sup _{0<\Vert a_0^{-1}b\Vert <r} \frac{\Vert  \phi (b) -\phi (a_0)- d\phi _{a_0}( a_0^{-1}b )\Vert }{\Vert  \phi (a_0)^{-1} a_0^{-1}b \phi (a_0)\Vert } =0.\) 
Analogously, \(\phi \)  is uniformly intrinsic differentiable in \(a_0\in \mathcal {O}\) , or \(\phi \)  is u.i.d. in \(a_0\in \mathcal {O}\) , if there is a intrinsic linear \(d\phi _{a_0}:\mathbb {W}\rightarrow \mathbb {H}\)  such that
\(\lim _{r \rightarrow 0^+}\sup _{a,b } \frac{ \Vert  \phi ( b) - \phi (a) - d\phi _{a_0}(a^{-1} b) \Vert }{\Vert \phi (a)^{-1}a^{-1}b\phi (a) \Vert } =0\) 
where \(r\)  is small enough so that \(\mathcal {U}(a_0,2r)\subset \mathcal {O}\)  and the supremum is for \(\Vert {a_0^{-1} a}\Vert <r,\, 0<\Vert {a^{-1} b}\Vert <r.\)
Finally, if \(k<m_1\)  is the dimension of \(\mathbb {H}\) , and if, without loss of generality, we assume that
\(\mathbb {H}=\lbrace p: p_{k+1}=\dots =p_N=0\rbrace \qquad \mathbb {W}=\lbrace p: p_{1}=\dots =p_k=0\rbrace \ \) 
then there is a \(k\times (m_1-k)\)  matrix, here denoted as \(\nabla ^\phi \phi (a_0)\) , such that
\(d\phi _{a_0} (b)= \left(\nabla ^\phi \phi (a_0) (b_{k+1},\dots ,b_{m_1})^T,0,\dots ,0\right),\) 
for all \(b=(b_1,\dots ,b_N)\in \mathbb {W}\) . The matrix \(\nabla ^\phi \phi (a_0)\)  is called the intrinsic horizontal Jacobian of \(\phi \)  in \(a_0\)  or the intrinsic horizontal gradient or even the intrinsic gradient if \(k=1\) .

Observe that u.i.d. functions do exist. In particular, when \(\mathbb {H}\)  is a horizontal subgroup, \(\mathbb {H}\)  valued euclidean 1 functions are u.i.d.
Theorem 2.12 (Theorem 4.7. in [10]}) 
If \(\mathbb {W}\)  and \(\mathbb {H}\)  are complementary subgroups of a Carnot group \(\mathbb {G}\)  with \(\mathbb {H}\)  horizontal and \(k\)  dimensional. If
\(\mathcal {O}\)  is open in \(\mathbb {W}\)  and \(\phi :\mathcal {O} \subset \mathbb {W}\rightarrow \mathbb {H}\)  is such that \(\phi \in 1( \mathcal {O}, \mathbb {H})\)  then \(\phi \)  is u.i.d. in \( \mathcal {O}\) .

In [10]}, the author gets a comparison between \(\mathbb {G}\) -regular surfaces (see Definition REF ) and the uniformly intrinsic differentiable maps:
Theorem 2.13 
Let \(\mathbb {W}\)  and \(\mathbb {H}\)  be complementary subgroups of a Carnot group \(\mathbb {G}\)  with \(\mathbb {H}\)  horizontal and \(k\)  dimensional.
Let \(\mathcal {O}\)  be open in \(\mathbb {W}\) , \(\phi :\mathcal {O} \subset \mathbb {W}\rightarrow \mathbb {H}\)  and \(S:= \mathrm {graph}\,(\phi )\) . Then the following are equivalent:

there are \( \mathcal {U}\)  open in \(\mathbb {G}\)  and \(f=(f_1,\dots , f_k)\in \mathbb {G}^1( \mathcal {U}; \mathbb {R}^k)\)  such that
\(\begin{split}& S=\lbrace p\in \mathcal {U}: f(p)=0\rbrace \\& d_{\bf P}f(q)_{\vert \mathbb {H}}:\mathbb {H}\rightarrow \mathbb {R}^k\quad \text{is bijective for all $q\in \mathcal {U}$}\end{split}\) 
and \(q\mapsto \left(d_{\bf P}f(q)_{\vert \mathbb {H}}\right)^{-1}\)  is continuous.

\(\phi \)  is u.i.d. in \(\mathcal {O}\) .

Moreover, if (1) or equivalently (2), hold then, for all \(a\in \mathcal {O}\)  the intrinsic differential \(d\phi _a\)  is
\(d\phi _a= - \left(d_{\bf P}f(a\phi (a))_{\vert \mathbb {H}} \right)^{-1}\circ d_{\bf P}f(a\phi (a))_{\vert \mathbb {W}}.\) 
Finally, if, without loss of generality, we choose a base \(X_1,\dots , X_N\)  of \(\mathfrak {g}\)  such that \(X_1,\dots , X_k\)  are horizontal vector fields, \(\mathbb {H}=\exp (\text{\rm span} \lbrace X_1,\dots , X_k\rbrace )\)  and \(\mathbb {W}=\exp (\text{\rm span} \lbrace X_{k+1},\dots , X_N\rbrace )\)  then
\(\mathbb {H}=\lbrace p: p_{k+1}=\dots =p_N=0\rbrace \qquad \mathbb {W}=\lbrace p: p_{1}=\dots =p_k=0\rbrace ,\) 
\(\nabla _\mathbb {G}f= \left(\, \mathcal {M}_1 \, \, | \,\, \mathcal {M}_2 \,\right)\) 
where
\(\mathcal {M}_1 := \begin{pmatrix}X_1f_1 \dots X_kf_1 \\\vdots \qquad \ddots \qquad \vdots \\X_1f_k \dots X_kf_k\end{pmatrix},\qquad \mathcal {M}_2 := \begin{pmatrix}X_{k+1}f_1\dots X_{m_1}f_1 \\\vdots \qquad \ddots \qquad \vdots \\X_{k+1}f_k \dots X_{m_1}f_k\end{pmatrix}.\) 
Finally, for all \(q\in \mathcal {U}\) , for all \(a\in \mathcal {O}\)  and for all \(p\in \mathbb {G}\)
\(\left(d_{\bf P}f(q)\right)(p)= \left(\nabla _\mathbb {G}f(q)\right)p^1\) 
and the intrinsic differential is
\(\begin{split}d\phi _a(b)&= \left( \left(\nabla ^\phi \phi (a)\right)(b_{k+1},\dots ,b_{m_1})^T,0 ,\dots , 0 \right)\\&= \left( \left( - \mathcal {M}_1(a\phi (a))^{-1}\mathcal {M}_2(a\phi (a))\right)(b_{k+1},\dots ,b_{m_1})^T,0 ,\dots , 0 \right),\end{split}\) 
for all \(b=(b_1,\dots ,b_N)\in \mathbb {W}\) .


Intrinsic Lipschitz Function
The following notion of intrinsic Lipschitz function appeared for the first time in [13]} and was studied, more diffusely, in [14]}, [15]}, [8]}, [4]}, [18]}, [19]}. Intrinsic Lipschitz functions play the same role as Lipschitz functions in Euclidean context.
Definition 2.9 
If \(\mathbb {W}, \mathbb {H}\)  are complementary subgroups in \(\mathbb {G}\) , \(q\in \mathbb {G}\)  and \(\beta \ge 0\) . We can define the cones \(C_{\mathbb {W},\mathbb {H}} (q,\beta ) \)  with base \(\mathbb {W}\)  and axis \(\mathbb {H}\) , vertex \(q\) , opening \(\beta \)  are given by
\(C_{\mathbb {W},\mathbb {H}} (q,\beta )=q\cdot C_{\mathbb {W},\mathbb {H}} (0,\beta )\) 
where \(C_{\mathbb {W},\mathbb {H}} (0,\beta )=\lbrace  p\, :\, \Vert p_\mathbb {W}\Vert  \le \beta \Vert p_\mathbb {H}\Vert  \rbrace \) .

For all \(\lambda >0\)  we have that \(\delta _\lambda (C_{\mathbb {W},\mathbb {H}} (0,\beta _1))=C_{\mathbb {W},\mathbb {H}} (0,\beta _1)\)  and if \(0<\beta _1<\beta _2\) , then
\(C_{\mathbb {W},\mathbb {H}} (q,\beta _1) \subset C_{\mathbb {W},\mathbb {H}} (q,\beta _2).\) 
Now we introduce the basic definitions of this paragraph.
Definition 2.10 
Let \(\mathbb {W}, \mathbb {H}\)  are complementary subgroups in \(\mathbb {G}\) . We say that \(\phi : \mathcal {O} \subset \mathbb {W}\rightarrow \mathbb {H}\)  is intrinsic \(C_L\) -Lipschitz in \(\mathcal {O}\)  for some \(C_L\ge 0\)  if for all \(C_1>C_L\)
\(C_{\mathbb {W},\mathbb {H}} (p, 1/C_1)\cap \mathrm {graph}\,(\phi ) =\lbrace p\rbrace  \quad \mbox{ for all } p\in \mathrm {graph}\,(\phi ).\) 
The Lipschitz constant of \(\phi \)  in \(\mathcal {O}\)  is the infimum of the \(C_1>0\)  such that (REF ) holds.
We will call a set \(S\subset \mathbb {G}\)  an intrinsic Lipschitz graph if there exists an intrinsic Lipschitz function \(\phi : \mathcal {O} \subset \mathbb {W}\rightarrow \mathbb {H}\)  such that \(S= \mathrm {graph}\,(\phi )\)  for suitable complementary subgroups \(\mathbb {W}\)  and \(\mathbb {H}\) .

We observe that the geometric definition of intrinsic Lipschitz graphs has equivalent analytic forms (see Proposition 3.1.3. in [4]}):
Proposition 2.14 
Let \(\mathbb {W}, \mathbb {H}\)  be complementary subgroups in \(\mathbb {G}\) , \(\phi : \mathcal {O} \subset \mathbb {W}\rightarrow \mathbb {H}\)  and \(C_L > 0\) . Then
the following statements are equivalent:

\(\phi \)  is intrinsic \(C_L\) -Lipschitz in \(\mathcal {O}\) .

\(\Vert  \mathbf {P}_\mathbb {H}( q^{-1} q^{\prime }) \Vert  \le C_L \Vert  \mathbf {P}_\mathbb {W}(q^{-1} q^{\prime }) \Vert  \, \, \)  for all \(q, q^{\prime } \in \mathrm {graph}\,(\phi ).\)

\(\Vert  \phi _{Q^{-1}} (a) \Vert  \le C_L \Vert  a \Vert  \, \, \)  for all \( q \in \mathrm {graph}\,(\phi )\)  and \(a\in \mathcal {O}_{ q^{-1}}\) .


If \(\phi : \mathcal {O} \subset \mathbb {W}\rightarrow \mathbb {H}\)  is intrinsic \(C_L\) -Lipschitz in \(\mathcal {O}\)  then it is continuous. Indeed if \(\phi (0)=0\)  then by the condition 3. of Proposition REF  \(\phi \)  is continuous in 0. To prove the continuity in \(a\in \mathcal {O}\) , observe that \(\phi _{q^{-1}}\)  is continuous in 0, where \(q=a\phi (a)\) .
Remark 2.15 
In this paper we are interested mainly in the special case when \(\mathbb {H}\)  is a horizontal subgroup and consequently \(\mathbb {W}\)  is a normal subgroup.
Under these assumptions, for all \(p= a\phi (a),q=b \phi (b) \in \mathrm {graph}\,(\phi ) \)  we have
\(\mathbf {P}_\mathbb {H}(p^{-1} q)= \phi (a)^{-1}\phi (b),\quad \mathbf {P}_\mathbb {W}(p^{-1} q)=\phi (a)^{-1} a^{-1}b\phi (a).\) 
Hence, if \(\mathbb {H}\)  is a horizontal subgroup, \(\phi : \mathcal {O} \subset \mathbb {W}\rightarrow \mathbb {H}\)  is intrinsic Lipschitz if
\(\Vert \phi (a)^{-1}\phi (b)\Vert  \le C_L \Vert \phi (a)^{-1} a^{-1}b\phi (a) \Vert  \qquad \text{for all $a,b \in \mathcal {O}$.}\) 
Moreover, if \(\phi \)  is intrinsic Lipschitz then \(\Vert \phi (a)^{-1} a^{-1}b\phi (a) \Vert \)  is comparable with \(\Vert {p^{-1} q}\Vert \) . Indeed from (REF )
\(\begin{split}c_0\Vert \phi (a)^{-1} a^{-1}b\phi (a) \Vert  &\le \Vert {p^{-1} q}\Vert \\ &\le \Vert \phi (a)^{-1} a^{-1}b\phi (a) \Vert +\Vert \phi (a)^{-1}\phi (b)\Vert \\&\le (1+C_L) \Vert \phi (a)^{-1} a^{-1}b\phi (a) \Vert .\end{split}\) 
The quantity \(\Vert \phi (a)^{-1} a^{-1}b\phi (a) \Vert \) , or better a symmetrized version of it, can play the role of a \(\phi \)  dependent, quasi distance on \(\mathcal {O}\) . See e.g. [21]}.

Remark 2.16 A map \(\phi \)  is intrinsic \(C_L\) -Lipschitz if and only if the distance of two points \(q, q^{\prime }\in \)  graph\((\phi )\)  is bounded by the norm of the projection of \(q^{-1} q^{\prime }\)  on the domain \(\mathcal {O}\) . Precisely \(\phi :\mathcal {O} \subset \mathbb {W}\rightarrow \mathbb {H}\)  is intrinsic \(C_L\) -Lipschitz in \(\mathcal {O}\)  if and only if there exists a constant \(C_1>0\)  satisfying
\(\Vert  q^{-1} q^{\prime } \Vert \le C_1\Vert  \mathbf {P}_\mathbb {W}(q^{-1} q^{\prime }) \Vert ,\) 
for all \(q,q^{\prime } \in \mathrm {graph}\,(\phi )\) . Moreover the relations between \(C_1\)  and the Lipschitz constant \(C_L\)  of \(\phi \)  follow from \((\ref {c_0})\) . In fact if \(\phi \)  is intrinsic \(C_L\) -Lipschitz in \(\mathcal {O} \)  then
\(\Vert  q^{-1} q^{\prime } \Vert \le \Vert  \mathbf {P}_\mathbb {W}(q^{-1} q^{\prime }) \Vert  +\Vert  \mathbf {P}_\mathbb {H}(q^{-1} q^{\prime }) \Vert  \le (1+C_L)\Vert  \mathbf {P}_\mathbb {W}(q^{-1} q^{\prime }) \Vert \) 
for all \(q,q^{\prime } \in \mathrm {graph}\,(\phi )\) . Conversely if \(\Vert  q^{-1} q^{\prime } \Vert \le c_0 (1+C_L)\Vert  \mathbf {P}_\mathbb {W}(q^{-1} q^{\prime }) \Vert \)  then
\(\Vert  \mathbf {P}_\mathbb {H}(q^{-1} q^{\prime } )\Vert \le C_L\Vert  \mathbf {P}_\mathbb {W}(q^{-1} q^{\prime }) \Vert \) 
for all \(q,q^{\prime } \in \mathrm {graph}\,(\phi )\) , i.e. the condition 2. of Proposition REF  holds.

We observe that in Euclidean spaces intrinsic Lipschitz maps are the same as Lipschitz maps. The converse is not true (see Example 2.3.9 in [8]}) and if \(\phi :\mathbb {W}\rightarrow \mathbb {H}\)  is intrinsic Lipschitz then this does not yield the existence of a constant \(C\)  such that
\(\Vert \phi (a)^{-1}\phi (b)\Vert \le C \Vert a^{-1}b\Vert  \quad \mbox{for } a,b\in \mathbb {W}\) 
not even locally. In Proposition 3.1.8 in [4]} the authors proved that the intrinsic Lipschitz functions, even if non metric Lipschitz, nevertheless are Hölder continuous.
Proposition 2.17 
Let \(\mathbb {W},\mathbb {H}\)  be complementary subgroups in \(\mathbb {G}\)  and \(\phi :\mathcal {O} \subset \mathbb {W}\rightarrow \mathbb {H}\)  be an intrinsic \(C_L\) -Lipschitz function. Then, for all \(r>0\) ,

there is \(C_1= C_1(\phi , r)>0\)  such that
\(\Vert \phi (a)\Vert  \le C_1 \quad \text{ for all $a\in \mathcal {O}$ with $\Vert a\Vert \le r$}\) 

there is \(C_2= C_2(C_L, r)>0\)  such that \(\phi \)  is locally \(1/\kappa \) -Hölder continuous i.e.
\(\Vert \phi (a)^{-1}\phi (b)\Vert \le C_2 \Vert a^{-1}b\Vert ^{1/\kappa }\quad \text{for all $a, b$ with $\Vert a\Vert  , \Vert b\Vert  \le r$}\) 
where \(\kappa \)  is the step of \(\mathbb {G}\) .


Now we present some results which we will use later:
Proposition 2.18 ([8]}, Theorem 4.2.9) 
If \(\hat{\phi }:\hat{ \mathcal {O}} \rightarrow  is intrinsic Lipschitz then the subgraph \) E\( of \)\( is a set with locally finite \) G\(-perimeter.\)
Proposition 2.19 (Proposition 3.6 [10]}) 
Let \(\mathbb {H}\) , \(\mathbb {W}\)  be complementary subgroups of \(\mathbb {G}\)  with \(\mathbb {H}\)  horizontal. Let \(\mathcal {O}\)  be open in \(\mathbb {W}\)  and \(\phi :\mathcal {O}\rightarrow \mathbb {H}\)  be u.i.d. in \(\mathcal {O}\) . Then

\(\phi \)  is intrinsic Lipschitz continuous in every relatively compact subset of \(\mathcal {O}\) .

the function \(a \mapsto d\phi _{a}\)  is continuous in \(\mathcal {O}\) .


Finally, we recall the following Rademacher type theorem, proved in [8]}, Theorem 4.3.5.
Theorem 2.20 
Let \(\mathbb {G}=\mathbb {W}\cdot \mathbb {H}\)  be a Carnot group of step 2 with \(\mathbb {H}\)  horizontal and one dimensional and let \(\hat{\phi }:\mathcal {O} \rightarrow \mathbb {H}\)  be intrinsic Lipschitz function, where \(\mathcal {O}\)  is a relatively open subset of \(\mathbb {W}\) . Then \(\hat{\phi }\)  is intrinsic differentiable \(( \mathcal {L}^{m+n-1}\mathop {\hbox{\vrule height 7pt width .5pt depth 0pt\vrule height .5pt width 6pt depth 0pt}}\nolimits \mathbb {W})\) -a.e. in \(\mathcal {O}\) . Notice that \( \mathcal {L}^{m+n-1} \mathop {\hbox{\vrule height 7pt width .5pt depth 0pt\vrule height .5pt width 6pt depth 0pt}}\nolimits \mathbb {W}\)  is the Haar measure of \(\mathbb {W}\) .

Carnot groups of step 2
In this section we present Carnot groups of step 2 as in Chapter 3 of [27]}. According to (REF ), the group operation of a Carnot group \(\mathbb {G}:= (\mathbb {R}^{m+n}, \cdot )\)  of step 2 is
\(p\cdot q= (p^1+q^1,p^2+q^2+\mathcal {Q}^2(p^1,q^1) )\) 
for every \(p=(p^1, p^2 )\) , \(q=(q^1,q^2 ) \in \mathbb {G}\) , where \(\mathcal {Q}^2=(\mathcal {Q}_1,\dots , \mathcal {Q}_n):\mathbb {R}^m\times \mathbb {R}^m \rightarrow \mathbb {R}^n\)  and by Proposition 2.2.22 \((4)\)  in [27]} we have
\(\mathcal {Q}_s (p^1,q^1)=\sum _{j,l=1}^m c^s_{jl} (p_jq_l-p_lq_j), \quad \mbox{ for } s=1,\dots , n\) 
with \( c_{jl}^{(s)}\in \mathbb {R}\)  for each \(j,l=1,\dots , m\) , \(s=1,\dots n\) . Hence if we consider \(n\)  skew-symmetric \(m\times m\)  real matrices \(\mathcal {B}^{(1)}, \dots , \mathcal {B}^{(n)}\) , it follows
\(\mathcal {Q}_s (p^1,q^1) = \langle \mathcal {B}^{(s)}p^1, q^1 \rangle , \quad \mbox{ for } s=1,\dots , n\) 
where \(\langle \cdot , \cdot \rangle \)  is the inner product in \(\mathbb {R}^m\) .
From Proposition 3.2.1 in [27]}, the authors explicitly remark that the linear independence of these matrices is also necessary for \(\mathbb {G}\)  to be a Carnot group. Consequently by also Theorem 3.2.2 in [27]}, we say that \(\mathbb {G}:= (\mathbb {R}^{m+n}, \cdot , \delta _\lambda )\)  is a Carnot group of step 2 if there are \(n\)  linearly independent, skew-symmetric \(m\times m\)  real matrices \(\mathcal {B}^{(1)}, \dots , \mathcal {B}^{(n)}\)  such that
for all \(p=(p^1,p^2)\)  and \(q= (q^1,q^2)\in \mathbb {R}^{m} \times \mathbb {R}^{n} \)  and for all \(\lambda >0\)
\(p\cdot q= (p^1+q^1 , p^2+q^2+ \frac{1}{2} \langle \mathcal {B}p^1,q^1 \rangle )\) 
where \(\langle \mathcal {B}p^1,q^1 \rangle := (\langle \mathcal {B}^{(1)}p^1,q^1 \rangle , \dots , \langle \mathcal {B}^{(n)}p^1,q^1 \rangle )\)  and \(\langle \cdot , \cdot \rangle \)  is the inner product in \(\mathbb {R}^m\)  and
\(\delta _\lambda p := (\lambda p^1 , \lambda ^2 p^2).\) 
We make the following choice of the homogeneous norm in \(\mathbb {G}\) :
\(\Vert (p^1,p^2)\Vert := \max \left\lbrace \vert p^1\vert _{\mathbb {R}^m}, \epsilon \vert p^2\vert _{\mathbb {R}^n}^{1/2} \right\rbrace \) 
for a suitable \(\epsilon \in (0,1]\)  (for the existence of such an \(\epsilon >0\)  see Theorem 5.1 in [3]}). We recall also that there is \(c_1>1\)  such that for all \(p=(p^1,p^2)\in \mathbb {G}\)
\(c_1^{-1}\left(\vert p^1\vert _{\mathbb {R}^m}+\vert p^2\vert _{\mathbb {R}^n}^{1/2} \right)\le \Vert p\Vert \le c_1\left(\vert p^1\vert _{\mathbb {R}^m}+\vert p^2\vert _{\mathbb {R}^n}^{1/2} \right)\) 
From now on we will depart slightly from the notations of the previous sections. Precisely, instead of writing \(p=(p_1,\dots , p_{m+n})\)  we will write
\(p=(x_1,\dots ,x_m,y_1,\dots , y_n).\) 
With this notation, when \(\mathcal {B}^{(s)}:=(b_{jl}^{(s)})_{ j,l=1}^m\) , a basis of the Lie algebra \(\mathfrak {g}\)  of \(\mathbb {G}\) , is given by the \(m+n\)  left invariant vector fields
\(X_j (p) = \partial _{x_j } +\frac{1}{2 } \sum _ {s=1 }^{n} \sum _ {l=1 }^{m} b_ {jl}^{(s)} x_l \partial _{y_s }, \qquad \qquad Y_s(p) = \partial _{y_s } ,\) 
where \(j=1,\dots ,m, \)  and \(s=1,\dots , n.\)
Remark 3.1 
The space of skew-symmetric \(m\times m\)  matrices has dimension \(\frac{m(m-1)}{2}\) . Hence in any Carnot group \(\mathbb {G}\)  of step 2 the dimensions \(m\)  of the horizontal layer and \(n\)  of the vertical layer are related by the inequality
\(n \le \frac{m(m-1)}{2}.\) 

Remark 3.2 Heisenberg groups \(\mathbb {H}^k= \mathbb {R}^{2k}\times \mathbb {R}\)  are Carnot groups of step 2 and
the group law is of the form \((\ref {1})\)  with
\(\mathcal {B}^{(1)}= \begin{pmatrix}0 & \mathcal {I}_k\\-\mathcal {I}_k & 0\end{pmatrix}\) 
where \(\mathcal {I}_k\)  is the \(k\times k\)  identity matrix.
More generally, H-type groups are examples of Carnot groups of step 2 (see Definition 3.6.1 and Remark 3.6.7 in [27]}). The composition law is of the form \((\ref {1})\)  where the matrices \(\mathcal {B}^{(1)},\dots , \mathcal {B}^{(n)}\)  have the following additional properties:

\( \mathcal {B}^{(s)}\)  is an \(m\times m \)  orthogonal matrix for all \(s=1,\dots n\)

\( \mathcal {B}^{(s)}\mathcal {B}^{(l)}=- \mathcal {B}^{(l)} \mathcal {B}^{(s)}\)  for every \(s,l=1,\dots ,n\)  with \(s\ne l\) .

Any H-type group is a H-group in the sense of Métivier, or a HM-group in short (see Section 3.7 in [27]}), which is also example of Carnot groups of step 2. Here the composition law is of the form \((\ref {1})\)  with the following additional condition: every non-vanishing linear combination of the matrices \( \mathcal {B}^{(s)}\) 's is non-singular.
Another example of Carnot groups of step 2 is provided by the class \(\mathbb {F} _{m,2}\)  of free groups of step-2 (see Section 3.3 in [27]}). Here \(\mathbb {F} _{m,2}= \mathbb {R}^{m}\times \mathbb {R}^{\frac{m(m-1)}{2}}\)  and the composition law \((\ref {1})\)  is defined by the matrices \(\mathcal {B}^{(s)}\equiv \mathcal {B}^{(l,j)}\)  where \(1\le j< l\le m\)  and \(\mathcal {B}^{(l,j)}\)  has entries \(-1\)  in position \((l,j)\) , 1 in position \((j,l)\)  and 0 everywhere else.
Notice that Heisenberg groups are H-type groups while \(\mathbb {H}^1\)  is also a free step-2 group.

The intrinsic gradient
Let \(\mathbb {G}= (\mathbb {R}^{m+n}, \cdot , \delta _\lambda )\)  be a Carnot group of step 2 as above and \(\mathbb {W}\) , \( be complementary subgroups in \) G\( with \)  horizontal and one dimensional.
Remark 3.3  To keep notations simpler,
through all this section we assume, without loss of generality, that the complementary subgroups \(\mathbb {W}\) , \( are\begin{equation}=\lbrace  (x_1,0\dots , 0) \rbrace , \qquad \mathbb {W}:=\lbrace  (0,x_2,\dots , x_{m+n}) \rbrace .\end{equation}This amounts simply to a linear change of variables in the first layer of the algebra \) g\(. If we denote \) M\( a non singular \) mm\( matrix, the linear change of coordinates associated to \) M\( is\begin{equation*}p=(p^1,p^2)\mapsto (\mathcal {M} p^1, p^2).\end{equation*}The new composition law \)\( in \) Rm+n\(, obtained by writing \)\( in the new coordinates, is\begin{equation*}(\mathcal {M} p^1, p^2) \star (\mathcal {M} q^1, q^2) := (\mathcal {M} p^1 +\mathcal {M} q^1, p^2 + q^2 + \frac{1}{2} \langle \mathcal { \tilde{B}} \mathcal {M} p^1 ,\mathcal {M} q^1 \rangle ),\end{equation*}where \) B := (B(1), ..., B(n))\(and\)   B (s)= (M-1)T B(s) M-1\( for \) s=1,...,n.\(It is easy to check that the matrices \)  B(1) ,..., B(n)\( are skew-symmetric and that \) (Rm+n, , )\( is a Carnot groups of step 2 isomorphic to \) G= (Rm+n, , )\(.\)
When \( and \) W\( are defined as in (\ref {5.2.0}) there is a natural inclusion \) i: Rm+n-1W\( such that, for all \) (x2,...xm,y1,...,yn)Rm+n-1\(,\)\(i((x_2,\dots x_m,y_1,\dots ,y_n)):=(0, x_2,\dots x_m,y_1,\dots ,y_n)\in \mathbb {W}.\)\(If \) O\( and \)\( are respectively an open set in \) Rm+n-1\( and a function \) :OR\(we denote \) O:= i(O)W\( and \) :O the function defined as
\(\hat{\phi }(i(a)) :=(\phi (a), 0,\dots ,0)\) 
for all \(a\in \mathcal {O}\) .
From (REF ) in Theorem \(\ref {teo4.1}\) , if \(\hat{\phi }:\hat{\mathcal {O}} \subset \mathbb {W}\rightarrow  is such that \) graph () \( islocally a non critical level set of \) f1G(G, R)\( with \) X1f0\(, then \)\( is u.i.d. in \) O\( and the following representation of the intrinsic gradient \)  \( holds\begin{equation}\nabla ^{\hat{\phi }} \hat{\phi }(p)=-\left(\frac{X_2f}{X_1f} ,\dots , \frac{X_{m}f}{X_1f}\right)(p\cdot \hat{\phi }(p))\end{equation}for all \) pO\(.\) In Proposition REF  we prove a different explicit expression of \(\nabla ^{\hat{\phi }} \hat{\phi }\) , not involving \(f\) , but only derivatives of the real valued function \(\phi \) .
Proposition 3.4 (Proposition 5.4, [10]}) 
Let \(\mathbb {G}:= (\mathbb {R}^{m+n}, \cdot , \delta _\lambda )\)  be a Carnot group of step 2 and \(, \) W\( the complementary subgroups defined in (\ref {5.2.0}). Let \) U\( be open in \) G\(, \) fC1G(U, R)\( with \) X1f>0\( and assume that \) S:={ pU : f(p)=0}\( is non empty. Then\begin{enumerate}\item [(i)]there are \hat{\mathcal {O}} open in \mathbb {W} and \hat{\phi } :\hat{\mathcal {O}} \rightarrow  such that S=\mathrm {graph}\,(\hat{\phi }) . Moreover \hat{\phi } is u.i.d. in \hat{\mathcal {O}} and the intrinsic gradient \nabla ^{\hat{\phi }} \hat{\phi } is the vector\nabla ^{\hat{\phi }} \hat{\phi }(i(a))=\left(D_2^{\phi } {\phi }(a),\dots , D_m^{\phi } {\phi }(a) \right)for all a\in \mathcal {O} where, for j=2,\dots , m,\begin{equation}D_j^{\phi } {\phi } = X_j \phi +\phi \sum _ {s=1 }^{n} b_ {j1 }^{(s)} Y_s\phi \end{equation}in distributional sense in \mathcal {O}, where X_j, Y_s are defined as (\ref {5.1.0}).\item [(ii)]The subgraph  \mathcal {E}:=\lbrace  p\in \mathcal {U} : f(p)<0\rbrace  has locally finite \mathbb {G}-perimeter in \mathcal {U} and its \mathbb {G}-perimeter measure |\partial \mathcal {E}|_\mathbb {G} has the integral representation|\partial \mathcal {E}|_\mathbb {G}(\mathcal {F}) =\int _{\Phi ^{-1}(\mathcal {F})} \sqrt{1+ |\nabla ^{\hat{\phi }} \hat{\phi }|_{\mathbb {R}^{m-1}}^2} \, \, d\mathcal {L}^{m+n-1}for every Borel set \mathcal {F} \subset \mathcal {U}  where \Phi :\mathcal {O}\rightarrow \mathbb {G} is defined as \Phi (a):=i(a)\cdot \hat{\phi }(i(a)) for all a\in \mathcal {O}.\end{enumerate}\)
From Proposition REF , if \(\mathrm {graph}\,(\phi )\)  is a \(\mathbb {G}\) -regular hypersurface, the intrinsic gradient of \(\phi \) 
takes the explicit form given in ().
This motivates the definitions of the operators intrinsic horizontal gradient and intrinsic derivatives.
Definition 3.1 
Let \(\mathcal {O} \)  be open in \(\mathbb {R}^{m+n-1}\) , \(\phi :\mathcal {O} \rightarrow \mathbb {R}\)  be continuous in \(\mathcal {O}\) . The intrinsic derivatives \(D^\phi _j \) , for \(j=2,\dots ,m\) , are the differential operators with continuous coefficients
\(\begin{split}D^\phi _j &:=\partial _{x_j }+\sum _ {s=1 }^{n} \left(\phi b_ {j 1}^{(s)} +\frac{1}{2 } \sum _ {i=2 }^{m} x_i b_ {ji }^{(s)} \right) \partial _{y_s } \\&= {X_j}_{\vert \mathbb {W}}+ \phi \,\sum _ {s=1 }^{n}b_ {j 1}^{(s)} {Y_s}_{\vert \mathbb {W}}\end{split}\) 
where, in the second line with abuse of notation, we denote with the same symbols \(X_j\)  and \(Y_s\)  the vector fields acting on functions defined in \(\mathcal {O}\) .
If \(\hat{\phi }:=(\phi , 0,\dots , 0):\hat{\mathcal {O}}\rightarrow , we denote\emph {intrinsic horizontal gradient} \)\( the differential operator\)\(\nabla ^{\hat{\phi }} :=(D^\phi _2,\dots , D^\phi _m).\)\(\)
Definition 3.2 \(\mathbf {(Distributional \, \, solution)}\) . Let \(\mathcal {O} \subset \mathbb {R}^{m+n-1}\)  be open and \(w=\left(w_2,\dots ,w_m\right)\in \mathcal {L}^\infty _{loc} (\mathcal {O} , \mathbb {R}^{m-1})\)  . We say that \(\phi \in \mathcal {O},\mathbb {R})\)  is a distributional solution in \(\mathcal {O}\)  of the non-linear first order PDEs' system
\(\left(D^{\phi }_2\phi ,\dots , D^{\phi }_m\phi \right)= w\) 
if for every \(\zeta \in 1 _c (\mathcal {O},\mathbb {R})\)
\(\int _\mathcal {O} \phi \left( \,X_j \zeta +\phi \sum _{s=1}^n b_ {j1 }^{(s)} Y_s \zeta \right) \, d\mathcal {L}^{m+n-1} = -\int _\mathcal {O} w_j \zeta \, d \mathcal {L}^{m+n-1}\) 
for \(j=2,\dots , m\) .

Remark 3.5 If the vector fields \(D^{\phi }_2,\dots , D^{\phi }_m \)  are smooth we know that it is possible to connect each couple of points \(a\)  and \(b\)  in \(\mathcal {O}\)  with a piecewise continuous integral curve of horizontal vector fields. This means that there is an absolutely continuous curve \(\gamma _h :[t_1,t_2]\rightarrow \mathcal {O} \)  from \(a\)  to \(b\)  such that \(-\infty <t_1<t_2<+\infty \)  and
\(\dot{\gamma }_h (t) = \sum _{j=2}^{m} h_j(t) \, D_j^\phi (\gamma _h (t)) \hspace{14.22636pt} \mbox{a.e.} \, \, t\in (t_1,t_2)\) 
with \(h=(h_2,\dots , h_m) :[t_1,t_2]\rightarrow \mathbb {R}^{m-1}\)  a piecewise continuous function. In our case the vector fields \(D_j^\phi \)  are only continuous, and consequently it isn't sure the existence of \(\gamma _h\) .

Proposition 3.6 
Let \(\mathbb {G}:= (\mathbb {R}^{m+n}, \cdot , \delta _\lambda )\)  be a Carnot group of step 2 and \(, \) W\( the complementary subgroups defined in (\ref {5.2.0}). Let \) :O  be intrinsic \(C_L\) -Lipschitz function, where \(\mathcal {O}\)  is open in \(\mathbb {W}\)  and \(\phi :\mathcal {O} \rightarrow \mathbb {R}\)  is the map associated to \(\hat{\phi }\)  as in \((\ref {phipsi})\) .
If \(\gamma :[t_1,t_2]\rightarrow \mathcal {O} \)  is an integral curve of the vector fields \(D^{\phi }_2,\dots , D^{\phi }_m \) , then \([t_1,t_2]\ni t \mapsto \phi (\gamma (t))\)  is Lipschitz continuous.

For simplicity we define \(\hat{\gamma }(t) := i(\gamma (t)) \in \hat{\mathcal {O}}\) . We would like to show that there exists \(C_1>0\)  such that
\(\Vert \hat{\phi }(\hat{\gamma }(t))^{-1}\hat{\gamma }(t)^{-1}\hat{\gamma }(t_1)\hat{\phi }(\hat{\gamma }(t)) \Vert  \le C_1 (t-t_1)\) 
for all \(t\in [t_1,t_2]\) . In fact since \(\hat{\phi }\)  is an intrinsic \(C_L\) -Lipschitz function and recall Remark REF  we have
\(|\phi (\gamma (t))- \phi (\gamma (t_1))|\le C_L \Vert \hat{\phi }(\hat{\gamma }(t))^{-1}\hat{\gamma }(t)^{-1}\hat{\gamma }(t_1)\hat{\phi }(\hat{\gamma }(t)) \Vert  \le C_L C_1 (t-t_1).\) 
By hypothesis \(\gamma = \gamma _h:[t_1,t_2]\rightarrow \mathcal {O} \)  is an absolutely continuous curve satisfying \((\ref {gamma})\)  with a piecewise continuous function \(h=(h_2,\dots , h_m) \in \mathcal {L}^\infty ((t_1,t_2),\mathbb {R}^{m-1})\)  and \(\gamma _h(t_1)=a=(x,y)\) , \(\gamma _h(t_2)=b=(x^{\prime },y^{\prime })\) .
More precisely we have \(\gamma _h(t)=(x_2(t),\dots , x_m(t),y_1(t),\dots ,y_n(t))=(x_\gamma (t),y_1(t),\dots ,y_n(t))\)  such that
\(x_l(t)-x_l=\int _{t_1}^t h_l(r)\, dr \hspace{11.38092pt} \mbox{ for all } \, t\in [t_1,t_2], \, l\in {2,\dots , m}\) 
\(y_s(t)-y_s= \sum _{j=2}^m \biggl (b_ {j1 }^{(s)}\int _{t_1}^t h_j(r)\phi (\gamma _h (r))\, dr + \frac{1}{2} \sum _{l=2}^m b_{jl}^{(s)} \biggl (x_l+ \int _{t_1}^t h_l(r) \, dr\biggl )\int _{t_1}^t h_j(r)\, dr \biggl )\) 
for all \( t\in (t_1,t_2)\) , \(s=1,\dots ,n\) .
Now we consider
\(\sigma _\phi (b,a) := \sum _{s=1}^{n} \Big |y_s-y^{\prime }_s+\phi (b) \Big (\sum _{l=2}^{m}(x_l -x^{\prime }_l)b_ {1l}^{(s)}\Big ) -\frac{1}{2} \langle \mathcal {B}^{(s)} x^{\prime }, x\rangle \Big |^{1/2}\) 
If we put \(\mathcal {B}_{M} = \max \lbrace  b_{ij}^{(s)} \, | \, i,j=1,\dots , m \, , s=1,\dots , n \rbrace \) , then it easy to see that
\(\sigma _\phi (b,a) \le \sigma _\phi (a,b) + n\sqrt{\mathcal {B}_M} |\phi (a)-\phi (b)|^{1/2} |x-x^{\prime }|^{1/2}_{\mathbb {R}^{m-1}}\) 
and, recalling (REF ), if \(b=\gamma _h(t)\)  then
\(\begin{aligned}\Vert \hat{\phi }(\hat{\gamma }(t))^{-1}\hat{\gamma }(t)^{-1}\hat{\gamma }(t_1)\hat{\phi }(\hat{\gamma }(t)) \Vert  & \le c_1\Big (|x-x_\gamma (t)|_{\mathbb {R}^{m-1}} +\sigma _\phi ( a, \gamma _h(t)) \\& + \, n\sqrt{\mathcal {B}_M}|x-x_\gamma (t)|_{\mathbb {R}^{m-1}}^{1/2}|\phi (\gamma _h(t))-\phi (a)|^{1/2} \Big ) .\end{aligned}\) 
By \(\hat{\phi }\)  is intrinsic \(C_L\) -Lipschitz, it follows
\(\begin{aligned}n\sqrt{\mathcal {B}_M}|x-x_\gamma (t)|_{\mathbb {R}^{m-1}}^{1/2} |\phi (\gamma _h(t))-\phi (a)|^{1/2} & \le n\sqrt{C_L\mathcal {B}_M}|x-x_\gamma (t)|_{\mathbb {R}^{m-1}}^{1/2} \Vert \hat{\phi }(\hat{\gamma }(t))^{-1}\hat{\gamma }(t)^{-1}\hat{\gamma }(t_1)\hat{\phi }(\hat{\gamma }(t)) \Vert  ^{1/2} \\& \le \frac{ n^2C_L\mathcal {B}_M}{2} |x-x_\gamma (t)|_{\mathbb {R}^{m-1}}+ \frac{1}{2} \Vert \hat{\phi }(\hat{\gamma }(t))^{-1}\hat{\gamma }(t)^{-1}\hat{\gamma }(t_1)\hat{\phi }(\hat{\gamma }(t)) \Vert  \\\end{aligned}\) 
for all \( t\in [t_1,t_2]\) . From \((\ref {x_i})\)
\(|x-x_\gamma (t)|_{\mathbb {R}^{m-1}} \le (m-1)(t-t_1)\Vert  h\Vert _{\mathcal {L}^\infty ((t_1,t_2),\mathbb {R}^{m-1})}\) 
and, consequently, if we put \(M_1:= c_1 (m-1)(2+C_L\mathcal {B}_Mn^2),\)  then
\(\Vert \hat{\phi }(\hat{\gamma }(t))^{-1}\hat{\gamma }(t)^{-1}\hat{\gamma }(t_1)\hat{\phi }(\hat{\gamma }(t)) \Vert  \le 2c_1\sigma _\phi (a,\gamma _h(t)) +M_1(t-t_1)\Vert  h\Vert _{\mathcal {L}^\infty ((t_1,t_2),\mathbb {R}^{m-1})}.\) 
Hence it remains to estimate \(\sigma _\phi (a,\gamma _h(t))\) .
By \((\ref {x_i})\)  and \((\ref {y_s})\)  we observe that
\(\begin{aligned}\sigma _\phi (a,\gamma _h(t)) & = \sum _{s=1}^n \biggl | \sum _{j=2}^m \left(b_ {j1 }^{(s)}\int _{t_1}^t h_j(r)\phi (\gamma _h (r))\, dr + \frac{1}{2} \sum _{l=2}^m b_{jl}^{(s)} \biggl (x_l+ \int _{t_1}^t h_l(r) \, dr\biggl ) \int _{t_1}^t h_j(r)\, dr \right)+ \\& \hspace{11.38092pt} +\phi (a)\sum _{l=2}^m b_{1l}^{(s)}\int _{t_1}^t h_l(r)\, dr -\frac{1}{2}\langle \mathcal {B}^{(s)} x, x_\gamma (t) \rangle \biggl |^{1/2}\end{aligned}\) 
with
\(\sum _{l,j=2}^m x_l b_{jl}^{(s)}\int _{t_1}^t h_j(r)\, dr - \langle \mathcal {B}^{(s)} x, x_\gamma (t) \rangle = \sum _{l,j=2}^m x_l b_{jl}^{(s)}(x_j(t)-x_j) - \langle \mathcal {B}^{(s)} x, x_\gamma (t)- x \rangle =0\) 
and
\(\biggl | \frac{1}{2} \sum _{l,j=2}^m b_{jl}^{(s)} \, \int _{t_1}^t h_l(r) \, dr \, \int _{t_1}^t h_j(r)\, dr \biggl | \le \frac{1}{2} \mathcal {B}_M (m-1)^2 (t-t_1)^2\Vert  h\Vert ^2_{\mathcal {L}^\infty ((t_1,t_2),\mathbb {R}^{m-1})}\) 
Hence, remembering that \(\phi \)  is intrinsic \(C_L\) -Lipschitz function, it follows
\(\begin{aligned}\sigma _\phi (a,\gamma _h(t)) & \le \sum _{s=1}^n \biggl | \sum _{j=2}^m b_{j1}^{(s)}\int _{t_1}^t h_j(r)\left(\phi (a)- \phi (\gamma _h (r))\right) \, dr \biggl |^{1/2} + \frac{\sqrt{2\mathcal {B}_M}}{2} n(m-1)(t-t_1)\Vert  h\Vert _{\mathcal {L}^\infty ((t_1,t_2),\mathbb {R}^{m-1})} \\& \le n \sqrt{\mathcal {B}_MC_L(m-1) (t-t_1)} \Vert  h\Vert ^{1/2}_{\mathcal {L}^\infty ((t_1,t_2),\mathbb {R}^{m-1})} \max _{ r\in [a,b]} \Vert \hat{\phi }(\hat{\gamma }(t))^{-1}\hat{\gamma }(t)^{-1}\hat{\gamma }(t_1)\hat{\phi }(\hat{\gamma }(t)) \Vert  ^{1/2}\\ & \quad + \frac{\sqrt{2\mathcal {B}_M}}{2} n(m-1)(t-t_1)\Vert  h\Vert _{\mathcal {L}^\infty ((t_1,t_2),\mathbb {R}^{m-1})}\end{aligned}\) 
Since \((\ref {stima1})\)  we conclude that
\(\begin{aligned}\max _{ t\in [t_1,t_2]} & \Vert \hat{\phi }(\hat{\gamma }(t))^{-1}\hat{\gamma }(t)^{-1}\hat{\gamma }(t_1)\hat{\phi }(\hat{\gamma }(t)) \Vert  \le M_1 (t-t_1) \Vert  h\Vert _{\mathcal {L}^\infty ((t_1,t_2),\mathbb {R}^{m-1})} \\ & \quad + 2c_1 n\sqrt{\mathcal {B}_M C_L(m-1) (t-t_1)} \Vert  h\Vert ^{1/2}_{\mathcal {L}^\infty ((t_1,t_2),\mathbb {R}^{m-1})} \max _{ t\in [t_1,t_2]}\Vert \hat{\phi }(\hat{\gamma }(t))^{-1}\hat{\gamma }(t)^{-1}\hat{\gamma }(t_1)\hat{\phi }(\hat{\gamma }(t)) \Vert  ^{1/2} \\& \quad +\sqrt{2\mathcal {B}_M} c_1n(m-1) (t-t_1) \Vert  h\Vert _{\mathcal {L}^\infty ((t_1,t_2),\mathbb {R}^{m-1})} \\& \le \biggl ( M_1+ 8 c_1^2n^2(m-1) \mathcal {B}_M C_L + \sqrt{2\mathcal {B}_M}c_1n(m-1) \biggl ) (t-a) \Vert  h\Vert _{\mathcal {L}^\infty ((t_1,t_2),\mathbb {R}^{m-1})} \\& \quad + \frac{1}{2} \max _{ t\in [t_1,t_2]} \Vert \hat{\phi }(\hat{\gamma }(t))^{-1}\hat{\gamma }(t)^{-1}\hat{\gamma }(t_1)\hat{\phi }(\hat{\gamma }(t)) \Vert  \\& =: \frac{M_2}{2} (t-t_1) \Vert  h\Vert _{\mathcal {L}^\infty ((t_1,t_2),\mathbb {R}^{m-1})} + \frac{1}{2} \max _{ t\in [t_1,t_2]} \Vert \hat{\phi }(\hat{\gamma }(t))^{-1}\hat{\gamma }(t)^{-1}\hat{\gamma }(a)\hat{\phi }(\hat{\gamma }(t)) \Vert  .\end{aligned}\) 
Consequently for all \(t\in [t_1,t_2]\)
\(\Vert \hat{\phi }(\hat{\gamma }(t))^{-1}\hat{\gamma }(t)^{-1}\hat{\gamma }(t_1)\hat{\phi }(\hat{\gamma }(t)) \Vert  \le M_2 (t-t_1) \Vert  h\Vert _{\mathcal {L}^\infty ((t_1,t_2),\mathbb {R}^{m-1})} =: C_1 (t-t_1),\) 
i.e. \((\ref {52cit})\)  holds.

Caccioppoli sets
Let \(\mathbb {G}\)  be a Carnot group of step 2. Let \(\mathbb {S}^{m-1}\)  be the unit sphere of \(\mathbb {R}^m\)  and \(\nu \in \mathbb {S}^{m-1}\) , i.e. \(\nu \in \mathbb {R}^m\)  and \(|\nu |_{\mathbb {R}^m}=1\) . By abuse of notation we identify \(\nu =(\nu _1,\dots , \nu _m) \in \mathbb {R}^m\)  and \(\nu =(\nu , 0 ,\dots , 0)\in \mathbb {G}\) .
Fix \(p=(p^1,p^2)\in \mathbb {G}\)  with \(p^1\in \mathbb {R}^{m}\) , \(p^2\in \mathbb {R}^n\) . Let \(\nu (p)=\langle p,\nu \rangle \nu \in \mathbb {G}\)  and we define \(\nu ^\perp (p)\in \mathbb {G}\)  as the unique point such that
\(p=\nu ^\perp (p) \cdot \nu (p)\) 
More precisely,
\(\nu ^\perp (p) = \biggl ( p^1-\langle p^1, \nu \rangle \nu , p^2- \frac{1}{2} \langle p^1, \nu \rangle \langle \mathcal {B}p^1, \nu \rangle \biggl ).\) 
We denote by \(\nu ^\perp = \lbrace  p=(p^1,p^2)\in \mathbb {G}\, :\, \langle p^1, \nu \rangle =0 \rbrace \)  the orthogonal complement of \(\nu \)  in \(\mathbb {G}\) . It is clear that \(\nu ^\perp (p)\in \nu ^\perp \)  for every \(p\in \mathbb {G}\) . It is clear that if \(\nu =(1,0,\dots ,0)\)  then
\( \nu ^\perp (p)= p_\mathbb {W}\qquad \nu (p)= p_\) 
where \(\mathbb {W}\) , \( are the complementary subgroups defined as (\ref {5.2.0}). Moreover according to Definition \ref {FMSdefi2.3.3}, the set\)\(\lbrace  q\in \mathbb {G}\, :\, \Vert  \nu ^\perp (p^{-1} q) \Vert  <\beta \Vert  \nu (p^{-1} q) \Vert  \rbrace \)\(is an intrinsic cone with vertex \) p\(, opening \) >0\( and axis specified by \)\(.\)
The main results of this section are Theorem REF  and Theorem REF . We show that the boundary of set with finite \(\mathbb {G}\) -perimeter and having a bound on the measure theoretic normal is an intrinsic graph of an intrinsic Lipschitz function. This is explained in Theorem 1.1 in [36]} in the context of Heisenberg groups.
Theorem 4.1 
Let \(E\subset \mathbb {G}\)  be a set with finite \(\mathbb {G}\) -perimeter in \(\mathcal {U}(0,r)\) , \(\nu _E\)  be the measure theoretic inward normal of \(E\)  and \(\nu \in \mathbb {S}^{m-1}\) . Assume that there exists \(k\in (0,1]\)  such that \(\langle \nu _E (p), \nu \rangle \le -k \)  for \(|\partial E|_\mathbb {G}\) -a.e. \(p\in \mathcal {U}(0,r)\) . Then there exists \(\beta >0\)  such that possibly modifying \(E\)  in a negligible set, we get for every \(p\in \partial E \cap \mathcal {U}(0,r)\)
\(\lbrace  q\in \mathcal {U}(0,r) \, :\, \Vert  \nu ^\perp (p^{-1} q) \Vert  <-\beta \langle p^{-1} q, \nu \rangle \rbrace  \subset E\) 
\(\,\quad \lbrace  q\in \mathcal {U}(0,r) \, :\, \Vert  \nu ^\perp (p^{-1} q) \Vert  <\beta \langle p^{-1} q, \nu \rangle \rbrace  \subset \mathbb {G}-E\) 

Theorem 4.2 
Under the same assumptions of Theorem \(\ref {teo11montivittone}\) , possibly modifying \(E\)  on a negligible set, the set \(\partial E \cap \mathcal {U}(0,r)\)  is the intrinsic graph of an intrinsic Lipschitz function.

Firstly we prove some preliminary results.
Lemma 4.3 
Let \(\mathbb {G}\)  be a Carnot group of step 2 and let \(E\subset \mathbb {G}\)  be a locally finite \(\mathbb {G}\) -perimeter set. Then for all \(\mathcal {U}(p,r)\)  with \(p\in \mathbb {G}\)  and \(r>0\) ,
if we consider a horizontal left invariant vector field \(Z\)  satisfying
\( \int _E Z\xi \, d \mathcal {L}^{m+n } \le 0 \quad \forall \xi \in 1_c (\mathcal {U}(p,r),\mathbb {R}),\, \xi \ge 0\) 
for each \(\mathcal {L}^{m+n}\) -measurable set \(F\subset \mathcal {U}(p,r)\) , we obtain
\(\mathcal {L}^{m+n}(E\cap F)\le \mathcal {L}^{m+n}(E\cap \exp tZ(F))\) 
for all \(t\ge 0\)  such that \(\exp t Z(F)\subset \mathcal {U}(p,r)\) .

Let \(\mathcal {U}(0,r)\)  with \(r>0\) . Notice that because the homogeneous norm is invariant we can assume \(p=0\) . Moreover thanks to Remark REF , without loss of generality, we also assume \(Z=X_1\) .
We consider the map \(\Theta :\mathbb {G}\rightarrow \mathbb {G}\)  given by
\(\Theta (p)=\exp p_1X_1(0,p_2,\dots , p_{m+n }).\) 
It is a global diffeomorphism and it satisfies
\(\det d\Theta (p)=1 \quad \mbox{ and } \quad \Theta _*\left( \partial _{ p_1} \right) =X_1.\) 
where \(d\Theta \)  denotes the differential of \(\Theta \) . If we put \(E_1:= \Theta ^{-1} (E)\)  and \(F_1:= \Theta ^{-1} (F)\) , then
\( \Theta (te_1+ F_1)= \exp (tX_1(F)), \quad t\in \mathbb {R}\) 
with \(e_1=(1,0,\dots , 0) \in \mathbb {R}^{m+n }\) . Moreover for all \(\theta \in 1_c (\Theta ^{-1}(\mathcal {U}(0,r)) , \mathbb {R})\)  with \(\theta \ge 0\)  we define \(\xi (q):=\theta (\Theta ^{-1} (q))\)  and consequently by (REF ) and (REF )
\(\int _{E_1} \partial _{ p_1} \theta (p)\, d \mathcal {L}^{m+n}(p)= \int _E X_1\xi (q)\, d \mathcal {L}^{m+n}(q) \le 0.\) 
Hence by Fubini-Tonelli Theorem and by a standard approximation argument we know that the function \(t \mapsto \chi _{E_1} (p+te_1) \)  is increasing for \(\mathcal {L}^{m+n}\) -a.e. \(p\in \Theta ^{-1}(\mathcal {U}(0,r))\)  as long as \(p+te_1 \in \Theta ^{-1}(\mathcal {U}(0,r))\) . Then for a certain \(t\ge 0\) , using again Fubini-Tonelli Theorem we obtain
\(\begin{aligned}\mathcal {L}^{m+n }(E_1\cap F_1) & =\int _{\mathbb {R}^{m+n-1} } \int _\mathbb {R}\chi _{E_1}(p) \chi _{F_1}(p) \, d \mathcal {L}d \mathcal {L}^{m+n -1}\\&\le \int _{\mathbb {R}^{m+n-1} } \int _\mathbb {R}\chi _{E_1}(te_1+p) \chi _{F_1}(p) \, d \mathcal {L}d \mathcal {L}^{m+n -1}\\&= \int _{\mathbb {R}^{m+n-1} } \int _\mathbb {R}\chi _{E_1}(p) \chi _{te_1+F_1}(p) \, d \mathcal {L}d \mathcal {L}^{m+n -1}\\&= \mathcal {L}^{m+n }(E_1\cap (te_1+F_1)).\end{aligned}\) 
Finally by the last inequality, \((\ref {mv218b})\)  and \((\ref {mv219c})\)  we obtain
\(\mathcal {L}^{m+n}(E\cap F)\le \mathcal {L}^{m+n}(E\cap \exp (tX_1(F))).\) 
Consequently the proof is complete.
Lemma 4.4 
Let \(\mathbb {G}\)  be a Carnot group of step 2. If \(k\in (0,1]\) , then there exists \(\beta =\beta (k)>0 \)  such that for all \(\nu \in \mathbb {S}^{m-1}\)  and \(p=(p^1,p^2)\in \mathbb {G}\)  satisfying
\(\Vert  \nu ^\perp (p)\Vert =\max \biggl \lbrace  |p^1 -\langle p^1, \nu \rangle \nu |_{\mathbb {R}^m} \, ,\,\epsilon | p^2- \frac{1}{2} \langle p^1, \nu \rangle \langle \mathcal {B}p^1, \nu \rangle |_{\mathbb {R}^n}^{1/2} \biggl \rbrace  \, \le - \beta \langle p^1, \nu \rangle ,\) 
there exist \(\eta _1, \dots , \eta _n \in \mathbb {R}^m\)  such that for all \(s=1,\dots , n\)
\(\begin{aligned}& \langle \eta _s, \nu \rangle \le -\sqrt{1-k^2}\, |\eta _s|_{\mathbb {R}^m} \qquad \langle p^1- \eta _s, \nu \rangle \le -\sqrt{1-k^2} \, | p^1- \eta _s|_{\mathbb {R}^m} \\& p^2= (\langle \mathcal {B}^{(1)} \eta _1, p^1 \rangle , \dots , \langle \mathcal {B}^{(n)} \eta _n, p^1 \rangle )\end{aligned}\) 

We split the proof of this lemma in two steps.
\(\mathbf {Step \, 1.}\)  First we show the case \(m=2\)  and \(n=1\) . Without loss of generality, we can assume that \(\nu =(1,0)\in \mathbb {S}^{1}\) . For \(h>0\)  and \(z=(z_1,z_2)\in \mathbb {R}^2\)  such that \(|z_2|\le -h z_1\)  consider the set
\(R_z(h)= \Big \lbrace  \eta =(\eta _1, \eta _2 ) \in \mathbb {R}^2 \, : \, |\eta _2 |\le -h\eta _1, \, \, |z_2-\eta _2 |\le -h (z_1-\eta _1) \Big \rbrace .\) 
For \(z=(z_1,z_2)\ne 0\) , the set \(R_z(h)\)  is a parallelogram with vertices 0, \(z\) ,
\(\hat{z}_1 =\biggl ( \frac{z_2 +hz_1}{2h}, \frac{z_2 +hz_1}{2} \biggl ) \qquad \mbox{and} \qquad \hat{z}_2 =\biggl ( \frac{hz_1-z_2}{2h}, \frac{z_2 -hz_1 }{2} \biggl ).\) 
The smooth function \(\varphi _z : R_z(h)\rightarrow \mathbb {R}\)  defined as
\(\varphi _z(\eta ):= \langle \mathcal {B}^{(1)}\eta , z \rangle = b_{12}(\eta _2 z_1 - \eta _1 z_2)\) 
is linear and attains the maximum and the minimum on \(\partial R_z (h)\) . If we assume that \( b_{12}>0\)  (for the other case \( b_{12}<0\)  it is
sufficient to exchange the roles of \(\hat{z}_1\)  and \(\hat{z}_2\) ), then
\(\max _{R_z(h)} \varphi _z = \varphi (\hat{z}_1)= b_{12} \frac{h^2z_1^2-z_2^2}{2h} \qquad \mbox{and} \qquad \min _{R_z(h)} \varphi _z = \varphi (\hat{z}_2)= b_{12} \frac{z_2^2-h^2z_1^2}{2h}\) 
Moreover if we consider the set
\(D_h = \biggl \lbrace  (z,y) \in \mathbb {R}^3 \, :\, |z_2|\le -h z_1, \, \, \frac{b_{12}}{2} \, \big (z_2^2-h^2z_1^2 \big ) \le yh\le \frac{b_{12}}{2} \, \big (h^2z_1^2-z_2^2\big ) \,\biggl \rbrace \) 
by continuity, \(\varphi _z\)  attains all the values between the maximum and the minimum. Hence for each \((z_1,z_2,y)\in D_h\)  there exists \(\eta \in R_z (h)\)  such that \(y= b_{12}(\eta _2 z_1 - \eta _1 z_2)\) .
Now let \(\beta >0\)  be a number satisfying
\(\beta \biggl (\frac{\beta }{\epsilon ^2} - \frac{b_{12}}{2} \biggl ) \le \frac{3b_{12} h }{8} \quad \mbox{with}\quad h=\sqrt{\frac{k^2}{2-k^2}}\) 
\(\beta ^2 \le \frac{k^2}{2-2k^2}\) 
From now on, \(h\)  is fixed and it depends on \(k\)  as in \((\ref {mv2.8p2}) \) .
Now we would like to show that each point \(P\in \mathbb {R}^3\)  satisfying \((\ref {mv2.6p22})\)  (with \(m=2\) , \(n=1\)  and \(\nu =(1,0)\) ) belongs to \(D_h\) .
Indeed let \(p=(p_1,p_2,p_3)\in \mathbb {R}^3\)  such that \((\ref {mv2.6p22})\)  is true, i.e.
\(\Vert  \nu ^\perp (p)\Vert =\max \lbrace  |p_2 | \, ,\, \epsilon | p_3- \frac{1}{2} b_{12} p_1 p_2 |^{1/2} \rbrace  \, \le -\beta p_1\) 
we have that
\(|p_2|\le - \beta p_1 \le - \frac{h}{2} p_1\) 
and
\(-\frac{\beta ^2 p_1^2 }{\epsilon ^2}+\frac{b_{12}}{2}p_1p_2 \le y\le \frac{\beta ^2 p_1^2 }{\epsilon ^2} +\frac{b_{12}}{2}p_1p_2.\) 
Consequently using \((\ref {mv2.8p2})\)  we deduce \(p_3 \le \frac{3}{8}b_{12} p_1^2 h \)  and
\(hp_3+\frac{b_{12}}{2} p_2^2 \le \frac{b_{12}}{2}p_1^2 h^2.\) 
The other estimate \(hp_3 - \frac{b_{12}}{2}p_2^2 \ge -\frac{b_{12}}{2} p_1^2 h^2\)  is obtained in the similar way. Hence \(p\in D_h\)  and so there exists \(\eta \in R_{p^1} (h)\)  such that \(p_3= \langle \mathcal {B}^{(1)}\eta , p^1 \rangle \) , where \(p^1=(p_1,p_2)\) .
Moreover by \((\ref {mv2.8p2})\) , \((\ref {mv2.9p2})\)  and \((\ref {mv2.6p22})\)  we obtain that
\(\begin{aligned}\langle \eta , \nu \rangle & \le -\frac{1}{ \sqrt{1+\beta ^2}}\, |\eta |\le -\sqrt{1-\frac{k^2}{2}}\, |\eta |\\ \langle p^1- \eta , \nu \rangle &\le -\sqrt{1-\frac{k^2}{2}} \, | p^1- \eta |,\end{aligned}\) 
i.e. (REF ) follows.
\(\mathbf {Step \, 2.}\)  We prove the proposition in the general case.
Let \( q^1, (q^{\prime })^1 \in \mathbb {R}^m -\lbrace 0\rbrace \)  with \(q^1\ne a(q^{\prime })^1\)  for all \(a \in \mathbb {R}- \lbrace  0 \rbrace \) . We denote \(p^1:=q^1+(q^{\prime })^1\)  and \( \mathcal {A}= \)  span\( \lbrace  q^1, (q^{\prime })^1 \rbrace  \)  and we consider the orthogonal projection of \(\nu \)  onto \(\mathcal {A}\)  by
\(\pi _{ p^1} \nu = \frac{\langle p^1, \nu \rangle p^1}{|p^1|_{\mathbb {R}^m}^2}\) 
If we put
\(\hat{\nu }=\frac{ \pi _{ p^1} \nu }{| \pi _{ p^1} \nu |} \qquad \mbox{and} \qquad \xi =| \pi _{ p^1} \nu |\) 
then \(\xi = \frac{|\langle p^1,\nu \rangle |}{|p^1|_{\mathbb {R}^m}}\)  and by \((\ref {mv2.6p22})\)  we conclude that
\(\frac{1}{\sqrt{1+\beta ^2}}\le \xi \le 1.\) 
Now we would like to show that if \((p^1,p^2)\)  satisfies \((\ref {mv2.6p22})\)  relatively to \(\nu \) , then \((p^1,\hat{p}^2)\)  with \( \hat{p}^2 = p^2/\xi ^2\)  satisfies \((\ref {mv2.6p22})\)  relatively to \(\hat{\nu }\)  with the same \(\beta \) . In fact, it is clear that
\(\epsilon |p^2 - \frac{1}{2} \langle p^1, \nu \rangle \langle \mathcal {B}^{(s)} p^1, \nu \rangle |_{\mathbb {R}^n}^{1/2} \le -\beta \langle p^1, \nu \rangle \qquad \Longleftrightarrow \qquad \epsilon | \hat{p}^2 - \frac{1}{2} \langle p^1, \hat{\nu }\rangle \langle \mathcal {B}^{(s)} p^1, \hat{\nu }\rangle |_{\mathbb {R}^n}^{1/2} \le -\beta \langle p^1, \hat{\nu }\rangle .\) 
Moreover, if \( |p^1|_{\mathbb {R}^m}\le -\sqrt{1+\beta ^2}\langle p^1,\nu \rangle \)  then
\(|p^1| _{\mathbb {R}^m}\le -\xi \sqrt{1+\beta ^2}\langle p^1,\hat{\nu }\rangle \le -\sqrt{1+\beta ^2}\langle p^1,\hat{\nu }\rangle \) 
and consequently \(|p^1-\langle p^1,\hat{\nu }\rangle \hat{\nu }|_{\mathbb {R}^m} \le -\beta \langle p^1,\hat{\nu }\rangle \) . By Step 1. we know that there exists \(\hat{\eta }_s \in \mathcal {A}\)  such that if \(p^2=(p_{m+1},\dots , p_{m+n})\) , then \(p_{m+s} = \langle \mathcal {B}^{(s)} \hat{\eta }_s, p^1 \rangle \)  where \(\eta _s = \xi ^2 \hat{\eta }_s\)  solves \(p_{m+s}= \langle \mathcal {B}^{(s)} \eta _s, p^1 \rangle \)  for \(s=1,\dots , n\) . Moreover
\(\langle \hat{\eta }_s, \hat{\nu }\rangle \le -\sqrt{1-\frac{k^2}{2}}\, | \hat{\eta }_s|_{\mathbb {R}^m} \qquad \mbox{and} \qquad \langle p^1 - \hat{\eta }_s, \hat{\nu }\rangle \le -\sqrt{1-\frac{k^2}{2}} \, | p^1- \hat{\eta }_s|_{\mathbb {R}^m}\) 
i.e. (REF ) follows.
Taking into account \((\ref {mv2.9p2})\) , \((\ref {mv2.13p2.2})\)  and \((\ref {mv2.16p2.2})\)  we get the first inequality in \((\ref {mv2.7p22})\) , indeed
\(\langle \eta _s, \nu \rangle = \xi \langle \eta _s,\hat{\nu }\rangle \le \xi ^3 \langle \hat{\eta }_s,\hat{\nu }\rangle \le -\xi |\eta _s|_{\mathbb {R}^m} \sqrt{1-\frac{k^2}{2}} \, \le - |\eta _s |_{\mathbb {R}^m}\frac{\sqrt{1-\frac{k^2}{2}}}{\sqrt{1+\beta ^2}} \, \le -|\eta _s|_{\mathbb {R}^m} \sqrt{1-k^2}\) 
for every \(s=1,\dots ,n\) . Finally using \((\ref {mv2.16p2.2})\)
\(\begin{aligned}\langle p^1- \eta _s, \nu \rangle & = \langle p^1- \xi ^2 \hat{\eta }_s, \nu \rangle = \xi ^2 \langle p^1- \hat{\eta }_s, \nu \rangle + (1-\xi ^2) \langle p^1, \nu \rangle = \xi ^3 \langle p^1- \hat{\eta }_s, \hat{\nu }\rangle + \xi (1-\xi ^2) \langle p^1, \hat{\nu }\rangle \\& \le \xi ^3 \biggl ( - \sqrt{1-\frac{k^2}{2} } \, \, | p^1 - \hat{\eta }_s|_{\mathbb {R}^m} \biggl ) + \xi (1-\xi ^2) \langle p^1,\hat{\nu }\rangle \end{aligned}\) 
and by \((\ref {mv2.13p2.2})\)  and \((\ref {mv2.14p2.2})\)
\(\begin{aligned}\langle p^1- \eta _s, \nu \rangle & \le \xi \biggl ( - \sqrt{1-\frac{k^2}{2} } \,\, |\xi ^2 p^1- \eta _s|_{\mathbb {R}^m} \biggl ) + (1-\xi ^2) \frac{-| p^1 |_{\mathbb {R}^m}}{\sqrt{1+\beta ^2}} \\& \le - \sqrt{1-k^2} \, \, |\xi ^2 p^1 - \eta _s|_{\mathbb {R}^m} - \frac{|(1-\xi ^2) p^1 |_{\mathbb {R}^m}}{\sqrt{1+\beta ^2}} \\& \le - \sqrt{1-k^2} \, \, |\xi ^2 p^1- \eta _s| _{\mathbb {R}^m} - |(1-\xi ^2) p^1|_{\mathbb {R}^m}\sqrt{1-k^2} \\& \le -\sqrt{1-k^2} \, | p^1- \hat{\eta }|_{\mathbb {R}^m}\end{aligned}\) 
i.e. the second inequality in \((\ref {mv2.7p22})\)  follows.
Now we are able to show the proof of Theorem REF . The following proof is based on the following observation: if we start from a point of \(E \cap \partial \mathcal {U}(0,r)\)  with positive lower density and we move for a short time along a horizontal direction near \(\nu \) , then we remain in the set of positive lower density of \(E\) . We can then show that for each point of \(E \cap \partial \mathcal {U}(0,r)\)  there is a truncated lateral cone with fixed opening that is contained in \(E\) . We use the similar technique exploited in Theorem 1.1 in [36]} in the context of Heisenberg groups.
Possibly modifying \(E\)  in a \(\mathcal {L}^{m+n}\) -negligible set, we can assume that \(E\)  coincides with the set of points where \(E\)  has positive lower density. Precisely
\(E= \biggl \lbrace  p\in \mathbb {G}\, :\, \liminf _{\delta \rightarrow 0} \frac{\mathcal {L}^{m+n}(E\cap \mathcal {U}_e(p,\delta ))}{\mathcal {L}^{m+n}(\mathcal {U}_e(p,\delta ))} >0 \biggl \rbrace \) 
where \(\mathcal {U}_e(p,\delta )\)  is the Euclidean ball centered at \(p\)  having radius \(\delta >0\) .
Let \(\beta =\beta (k)>0\)  as in Lemma \(\ref {mvprop2.2}\) . We would like to show that for every \(p\in E\)
\(\lbrace  q\in \mathcal {U}(0,r) \, :\, \Vert  \nu ^\perp (p^{-1} q) \Vert  <-\beta \langle p^{-1} q, \nu \rangle \rbrace  \subset E\) 
First we define
\(\begin{aligned}\mathcal {A}_1(p)& :=\lbrace  \exp t Z_\mu (p) \in \mathcal {U}(0,r) \, :\, t\ge 0, \mu \in \mathbb {S}^{m-1}_k \rbrace \\\end{aligned}\) 
where \( \mathbb {S}^{m-1}_k:=\lbrace  \mu \in \mathbb {S}^{m-1} : \langle \mu , \nu \rangle \le - \sqrt{1-k^2} \, \rbrace  \)  and \(Z_\mu \)  is the left invariant vector field
\(Z_\mu =\mu _1 X_1 + \dots +\mu _{m} X_m, \quad \mbox{ for } \mu =(\mu _1, \dots , \mu _m)\in \mathbb {S}^{m-1}_k.\) 
For any \(\xi \in 1_c (\mathcal {U}(0,r) , \mathbb {R})\)  such that \(\xi \ge 0\)  and for all \(\mu \in \mathbb {S}^{m-1}_k\)  it follows
\(\int _E Z_\mu \xi \, d\mathcal {L}^{m+n}=- \int _{\mathcal {U}(0,r)} \xi \langle \mu , \nu _E \rangle \, d |\partial E|_\mathbb {G}\le 0.\) 
Indeed we know that \(\langle \mu , \nu _E (p)\rangle \ge 0\)  for \(|\partial E |_\mathbb {G}\) -a.e. \(p\in \mathcal {U}(0,r)\)  because \(\langle \mu , \nu \rangle \le - \sqrt{1-k^2}\)  and \(\langle \nu _E , \nu \rangle \le -k\) . Then using Lemma \(\ref {mvlemma2.1}\)  we conclude that if \(p\in E\cap \mathcal {U}(0,r)\) , \(t>0\)  is such that \(\exp tZ_\mu (p)\in \mathcal {U}(0,r)\)  and \(\delta >0\)  is small enough, then
\(\mathcal {L}^{m+n} (E\cap \exp tZ_\mu (\mathcal {U}_e(p,\delta ))) \ge \mathcal {L}^{m+n} (E\cap \mathcal {U}_e(p,\delta )).\) 
Moreover by \(\mathcal {L}^{m+n} ( \exp tZ_\mu (\mathcal {U}_e(p,\delta ))) = \mathcal {L}^{m+n} ( \mathcal {U}_e(p,\delta ))\)  we deduce
\(\liminf _{\delta \rightarrow 0} \frac{\mathcal {L}^{m+n}(E\cap \exp tZ_\mu (\mathcal {U}_e(p,\delta )))}{\mathcal {L}^{m+n}(\exp tZ_\mu (\mathcal {U}_e(p,\delta )))} \ge \liminf _{\hat{r} \rightarrow 0} \frac{\mathcal {L}^{m+n}(E\cap \mathcal {U}_e(p,\delta ))}{\mathcal {L}^{m+n}(\mathcal {U}_e(p,\delta ))} >0\) 
and consequently the point \(q=\exp tZ_\mu (p)\)  satisfies
\(\liminf _{\delta \rightarrow 0} \frac{\mathcal {L}^{m+n}(E\cap \mathcal {U}_e(q,\delta ))}{\mathcal {L}^{m+n}(\mathcal {U}_e(q,\delta ))} >0.\) 
This implies that \(\exp tZ_\mu (p) \in E\)  and \(\mathcal {A}_1(p) \subset E\) . Now if \(p=0 \in E\) , then
\(\begin{aligned}\mathcal {A}_1(0)& =\lbrace  \exp t Z_\mu (0) \in \mathcal {U}(0,r) \, :\, t\ge 0, \mu \in \mathbb {S}^{m-1}_k \rbrace \\& = \lbrace  (\eta , 0 )\in \mathbb {G}\, :\, \eta \in \mathbb {R}^m, \, \langle \eta , \nu \rangle \le -|\eta |\sqrt{1-k^2}, \, |\eta |<r \rbrace .\end{aligned}\) 
Moreover if we consider the conditions \((\ref {mv2.7p22})\)  of Lemma \(\ref {mvprop2.2}\)  and we define
\(\begin{aligned}\mathcal {A}_2 & :=\lbrace  \exp t Z_\mu (\eta , 0) \in \mathcal {U}(0,r) \, :\, t\ge 0, \mu \in \mathbb {S}^{m-1}_k, \, (\eta , 0)\in \mathcal {A}_1(0) \rbrace \\\end{aligned}\) 
then
\(\begin{aligned}\mathcal {A}_2 & = \lbrace  (p^1,p^2 )\in \mathbb {G}\, : \mbox{ there are } \eta _1, \dots ,\eta _n \in \mathbb {R}^m, \, |\eta _s|_{\mathbb {R}^m}<r \mbox{ such that } (\ref {mv2.7p22}) \mbox{ holds } \rbrace \end{aligned}\) 
and \(\mathcal {A}_2\subset E\) . Hence by Proposition \(\ref {mvprop2.2}\)  we obtain
\(\lbrace  q\in \mathcal {U}(0,r) \, :\, \Vert  \nu ^\perp (q) \Vert  <-\beta \langle q, \nu \rangle \rbrace  \subset \mathcal {A}_2 \subset E\) 
i.e. (REF ) is true in the case \(p=0\) . Precisely (REF ) follows for all for each \(p\in E\)  from the case \(p=0\)  by a left translation.
Now we consider \(\mathbb {G}-E\)  where \(\nu _{\mathbb {G}-E }=-\nu _E \)  in \(\mathcal {U}(0,r)\) . We can repeat the previous argument and we obtain for each \(P\)  where \(\mathbb {G}-E\)  has positive lower density,
\(\lbrace  q\in \mathcal {U}(0,r) \, :\, \Vert  \nu ^\perp (p^{-1} q) \Vert  <\beta \langle p^{-1} q, \nu \rangle \rbrace  \subset \mathbb {G}-E\) 
Precisely (REF ) holds for all \(p\in \mathcal {U}(0,r)-E\)  because \(\mathbb {G}-E\)  has density 1 at such \(p\) .
Approximating a point \(p\in \partial E \cap \mathcal {U}(0,r) \)  with a sequence of points in \(E\cap \mathcal {U}(0,r)\) , from (REF ) we obtain (REF ). Moreover approximating a point \(p \)  with a sequence of points in \(\mathcal {U}(0,r) - E\) , using (REF ), (REF ) holds. Possibly we have to take a smaller \(\beta \) .
Finally we are able to show the proof of Theorem REF .
Let \(\mathbf {P}_{ \nu ^\perp } :\mathbb {G}\rightarrow \nu ^\perp \)  be the projection map. By (REF ) we have that \(\mathbf {P}_{ \nu ^\perp } (E\cap \mathcal {U}(0,r))\)  is open set in \(\mathbf {P}_{ \nu ^\perp } ( \mathcal {U}(0,r))\)  and relatively open in \( \nu ^\perp \) . Let
\(\hat{ \mathcal {O}} := \lbrace  p\in \mathbf {P}_{ \nu ^\perp } (E\cap \mathcal {U}(0,r)) \,:\, \mbox{there is } t\in \mathbb {R}\mbox{ such that } \exp tZ_\nu (p) \in \mathcal {U}(0,r) -E\rbrace .\) 
By (REF ) and (REF ) we deduce that \(\hat{ \mathcal {O}}\)  is relatively open in \(\mathbf {P}_{ \nu ^\perp } ( \mathcal {U}(0,r))\)  and so in \( \nu ^\perp \) . Then from Theorem REF , the function \(\hat{\phi }= \phi \nu :\hat{ \mathcal {O}} \rightarrow \mathbb {G}\)  defined as
\(\phi (p):=\sup \lbrace t\in \mathbb {R}\, :\, \exp tZ_\nu (p) \in \mathcal {U}(0,r) \mbox{ and } \chi _E ( \exp tZ_\nu (p)) =1 \rbrace ,\) 
is intrinsic Lipschitz map because \(\mathrm {graph}\,(\hat{\phi }) \cap \lbrace  Q\in \mathcal {U}(0,r) \, :\, \Vert  \nu ^\perp (p^{-1} q) \Vert  <\beta \langle p^{-1} q, \nu \rangle \rbrace  = \emptyset \)  (see Definition REF ) and
\(\partial E\cap \mathcal {U}(0,r)=\lbrace  p\cdot \hat{\phi }(p) \,:\, p\in \mathcal {O}\rbrace ,\) 
i.e. the thesis is true.

area formula in carnot groups of step 2
In this section we prove an area formula beside the spherical Hausdorff measure for the graph of an intrinsic Lipschitz function (see Theorem REF ) and that the
pointwise gradient coincides with the weak one (see Proposition ). We observe that this fact is not elementary at all in our situation, since the \(\nabla ^{\hat{\phi }} \)  is not well defined when \(\hat{\phi }\)  is an intrinsic Lipschitz function.
In this section we examine a Carnot group \(\mathbb {G}\)  of step 2 and \(, \) W\( are the complementary subgroups defined in (\ref {5.2.0}).\) Let \(\hat{\phi }:\hat{ \mathcal {O}} \rightarrow  be intrinsic Lipschitz function, where \)  O\( is an open in \) W\( and \) :OR\( is the map associated to \)\( as in \) (REF )\(. We recall that \) :O G\( is the map graph of \)\( defined as \)\(\Phi (a):=i(a) \cdot \hat{\phi }(i(a)).\)\(Moreover we define the intrinsic subgraph of \)\( as\begin{equation}\mathcal {E}=\mathcal {E}_{\hat{\phi }} := \lbrace i(a)\cdot (t,0,\dots , 0 )\in \hat{ \mathcal {O}} \cdot :\, t < \phi (a) \rbrace .\end{equation}\) We begin with a result about the intrinsic generalized inward normal \(\nu _{\mathcal {E}}\)  to the intrinsic subgraph:
Lemma 5.1 
Let \(\mathbb {G}:= (\mathbb {R}^{m+n}, \cdot , \delta _\lambda )\)  be a Carnot group of step 2 and \(, \) W\( the complementary subgroups defined in (\ref {5.2.0}). Let \) : O  be an intrinsic Lipschitz function, where \(\hat{ \mathcal {O}}\)  is an open subset of \(\mathbb {W}\)  and \(\phi :\mathcal {O} \rightarrow \mathbb {R}\)  is the map associated to \(\hat{\phi }\)  as in \((\ref {phipsi})\) .
Then the intrinsic generalized inward normal \(\nu _{\mathcal {E}}\)  to the intrinsic subgraph \(\mathcal {E}\)  in \(\mathbb {G}\)  has the following representation
\(\nu _{\mathcal {E}}\left(\Phi (a)\right)=\left( - \frac{1}{\sqrt{1+|\nabla ^{\hat{\phi }} \hat{\phi }(i(a)) |_{\mathbb {R}^{m-1}}^2}} ,\frac{\nabla ^{\hat{\phi }} \hat{\phi }(i(a))}{\sqrt{1+|\nabla ^{\hat{\phi }} \hat{\phi }(i(a))|_{\mathbb {R}^{m-1}}^2}} \right)\) 
for a.e. \(a\in \mathcal {O}\) .

By Theorem REF , Theorem REF  and Theorem REF  we have that \(\hat{\phi }\)  is intrinsic differentiable a.e. in \(\hat{ \mathcal {O}}\)  and for all \(i(a)\in \hat{ \mathcal {O}} \)  point of intrinsic differentiability of \(\hat{\phi }\) , there exists a unique \(d\hat{\phi }_{i(a)}:\mathbb {W}\rightarrow  intrinsic differential of \)\( at \) i(a)\( such that\begin{equation*}\mathrm {graph}\,(d\hat{\phi }_{i(a)})=\Big \lbrace  (p_1,\dots , p_{m+n}) \in \mathbb {G}\, : \, \sum _{j=1}^{m} \nu _{\mathcal {E}}^{(j)} (\Phi (a)) p_j =0 \Big \rbrace \end{equation*}where \) E(1),..., E(m)\( are the components of \) E\(. Then we obtain\begin{equation*}\begin{aligned}\left\lbrace  b\cdot d\hat{\phi }_{i(a)}(b) \in \mathbb {G}\, :\, b\in \mathbb {W}\right\rbrace  =\Big \lbrace  (p_1,\dots , p_{m+n}) \in \mathbb {G}\, : \, \sum _{j=1}^{m} \nu _{\mathcal {E}}^{(j)} (\Phi (a)) p_j =0 \Big \rbrace .\end{aligned}\end{equation*}By \) (REF )\( there is \) ( D2  (a), ..., Dm  (a) )Rm-1\( associated to \) di(a)\( such that\)\(d\hat{\phi }_{i(a)}(b)= \Big ( \sum _{j=2}^{m} D_j^{\phi } \phi (a) x_j ,0,\dots , 0\Big ),\)\(for all \) b=(0,x2,..., xm, y1,..., yn)W\(, and consequently recalling that \) (bdi(a)(b))1=( j=2m Dj (a) xj, x2,...,xm )\( we deduce\begin{equation*}\nu _{\mathcal {E}}^{(1)} (\Phi (a)) \sum _{j=2}^{m} D_j^{\phi } \phi (a) x_j + \sum _{j=2}^{m} \nu _{\mathcal {E}}^{(j)} (\Phi (a)) x_j =0\end{equation*}for all \) b=(0,x2,..., xm, y1,..., yn)W\(. The thesis follows choosing \) b=(0,..., 0 ,1,0,..., 0)\( where \) j\(-th element is \) 1\( for \) j=2,...,m\( and recalling that \) | E((a))|Rm=1\( a.e. in \) O \(.\)
Lemma 5.2 ([8]}, Lemma 4.2.10) 
Let \(\hat{\phi }:\hat{ \mathcal {O}} \rightarrow  be an intrinsic Lipschitz function. Then there exists \) C(W, >0\( such that\begin{equation*}(\Phi )_* (\mathcal {L}^{m+n-1} \mathop {\hbox{\vrule height 7pt width .5pt depth 0pt\vrule height .5pt width 6pt depth 0pt}}\nolimits \mathbb {W})=- C(\mathbb {W},  \nu _{\mathcal {E}}^{(1)} |\partial \mathcal {E}|_\mathbb {G}\end{equation*}where \) ()* (Lm+n-1 W)\( denotes the image of \) Lm+n-1 W\( under the map \)\( and \) E(1)\( is the first component of the intrinsic generalized inward normal to the intrinsic subgraph \) E\(. \)
Proposition 5.3 
Let \(\mathbb {G}:= (\mathbb {R}^{m+n}, \cdot , \delta _\lambda )\)  be a Carnot group of step 2 and \(, \) W\( the complementary subgroups defined in (\ref {5.2.0}). Let \) : O  be an intrinsic Lipschitz function, where \(\hat{ \mathcal {O}}\)  is an open subset of \(\mathbb {W}\)  and \(\phi :\mathcal {O} \rightarrow \mathbb {R}\)  is the map associated to \(\hat{\phi }\)  as in \((\ref {phipsi})\) .
Then for every \(\xi =(\xi _1,\dots , \xi _m) \in 1_c (\hat{ \mathcal {O}} \cdot  \mathbb {R}^m)\)  we have
\(\int _{\mathcal {E}} \mbox{div} _\mathbb {G}\xi \, d\mathcal {L}^{m+n} = \frac{1}{C(\mathbb {W},} \int _\mathcal {O} \xi _1 \circ \Phi - \sum _{j=2}^{m} D_j^{\phi } \phi ( \xi \circ \Phi )_j \, d\mathcal {L}^{m+n-1}\) 
where \(C(\mathbb {W},>0\)  is given by Lemma \(\ref {coro4.3}\)  and \(\Phi :\mathcal {O} \rightarrow \mathbb {G}\)  is the map graph of \(\hat{\phi }\) .

Thanks to Proposition REF  we know that \(\mathcal {E}\)  is a set of locally finite perimeter in \(\mathbb {G}\)  and consequently by Structure Theorem of BV\(_\mathbb {G}\)  functions (see Theorem REF ) there exists a unique \(|\partial \mathcal {E}|_\mathbb {G}\) -measurable function \(\nu _\mathcal {E}:\hat{ \mathcal {O}} \cdot \mathbb {R}^m\)  such that \(|\nu _\mathcal {E}|_{\mathbb {R}^{m}}=1\)  \(|\partial \mathcal {E}|_\mathbb {G}\) -a.e. in \(\hat{ \mathcal {O}} \cdot  and\begin{equation}\int _{\mathcal {E}} \mbox{div} _\mathbb {G}\xi \, d\mathcal {L}^{m+n} = - \int _{\hat{ \mathcal {O}} \cdot  \langle \xi ,\nu _\mathcal {E}\rangle \, d|\partial \mathcal {E}|_\mathbb {G}}for all \xi \in 1_c (\hat{ \mathcal {O}} \cdot  \mathbb {R}^m) with |\xi |_{\mathbb {R}^{m}}\le 1.Now using the Lemma \ref {coro4.2} we have that the first component \nu _\mathcal {E}^{(1)} of \nu _\mathcal {E} is such that \nu _\mathcal {E}^{(1)}\ne 0 |\partial \mathcal {E}|_\mathbb {G}-a.e. in \hat{ \mathcal {O}} \cdot . Moreover from Lemma \ref {coro4.3} there exists C(\mathbb {W},>0 such that\begin{equation*}-\int _{\hat{ \mathcal {O}} \cdot  \langle \xi ,\nu _\mathcal {E}\rangle \, d|\partial \mathcal {E}|_\mathbb {G}=- \int _{\hat{ \mathcal {O}} \cdot  \frac{\langle \xi ,\nu _\mathcal {E}\rangle }{\nu _\mathcal {E}^{(1)}}\, \nu _\mathcal {E}^{(1)} \, d|\partial \mathcal {E}|_\mathbb {G}= \frac{1}{C(\mathbb {W},} \int _{\hat{ \mathcal {O}} \cdot  \frac{\langle \xi ,\nu _\mathcal {E}\rangle }{\nu _\mathcal {E}^{(1)}}\, d\Phi _* (\mathcal {L}^{m+n-1} \mathop {\hbox{\vrule height 7pt width .5pt depth 0pt\vrule height .5pt width 6pt depth 0pt}}\nolimits \mathbb {W}).}Finally since a change of variables we conclude that\begin{equation*}\frac{1}{C(\mathbb {W},} \int _{\hat{ \mathcal {O}} \cdot  \frac{ \langle \xi ,\nu _\mathcal {E}\rangle }{\nu _\mathcal {E}^{(1)}}\, d\Phi _* (\mathcal {L}^{m+n-1} \mathop {\hbox{\vrule height 7pt width .5pt depth 0pt\vrule height .5pt width 6pt depth 0pt}}\nolimits \mathbb {W}) = \frac{1}{C(\mathbb {W},} \int _\mathcal {O} \frac{\langle \nu _\mathcal {E}\circ \Phi , \xi \circ \Phi \rangle }{\nu _\mathcal {E}^{(1)} \circ \Phi } \, d\mathcal {L}^{m+n-1}}and consequently, by (\ref {63cit}) for all \xi =(\xi _1,\dots , \xi _m)\in 1_c (\hat{ \mathcal {O}} \cdot  \mathbb {R}^m) with |\xi |_{\mathbb {R}^{m}}\le 1\begin{equation}\begin{aligned}\int _{\mathcal {E}} \mbox{div} _\mathbb {G}\xi \, d\mathcal {L}^{m+n} & = -\int _{\hat{ \mathcal {O}} \cdot  \langle \xi ,\nu _\mathcal {E}\rangle \, d|\partial \mathcal {E}|_\mathbb {G}\\& = \frac{1}{C(\mathbb {W},} \int _\mathcal {O} \frac{\langle \nu _\mathcal {E}\circ \Phi , \xi \circ \Phi \rangle }{\nu _\mathcal {E}^{(1)} \circ \Phi } \, d\mathcal {L}^{m+n-1}\\& = \frac{1}{C(\mathbb {W},} \int _\mathcal {O} \biggl ( \xi _1 \circ \Phi + \sum _{j=2}^{m} \frac{(\nu _\mathcal {E}\circ \Phi )_j}{\nu _\mathcal {E}^{(1)} \circ \Phi } \, (\xi \circ \Phi )_j \biggl )\, d\mathcal {L}^{m+n-1}\\& = \frac{1}{C(\mathbb {W},} \int _\mathcal {O} \biggl ( \xi _1 \circ \Phi - \sum _{j=2}^{m} D_j^{\phi } \phi ( \xi \circ \Phi )_j \biggl )\, d\mathcal {L}^{m+n-1}}\end{aligned}Hence putting together the last equality and (\ref {eqDIV}) we obtain the thesis.\end{equation}\end{equation*}\begin{prop}Let \mathbb {G}:= (\mathbb {R}^{m+n}, \cdot , \delta _\lambda ) be a Carnot group of step 2 and , \mathbb {W} the complementary subgroups defined in (\ref {5.2.0}). Let \hat{\phi }:\hat{ \mathcal {O}} \rightarrow  be a locally intrinsic Lipschitz function, where \hat{ \mathcal {O}} is an open subset of \mathbb {W} and \phi :\mathcal {O} \rightarrow \mathbb {R} is the map associated to \hat{\phi } as in (\ref {phipsi}).Then the intrinsic gradient (D_2^\phi \phi ,\dots ,D_m^\phi \phi ) is also distributional, i.e.\begin{equation*}\int _\mathcal {O} \phi \left( \,X_j \zeta +\phi \sum _{s=1}^n b_ {j1 }^{(s)} Y_s \zeta \right) \, d\mathcal {L}^{m+n-1} =- \int _\mathcal {O} D^\phi _j \phi \, \zeta \, d \mathcal {L}^{m+n-1}\end{equation*}for every \zeta \in 1_c (\mathcal {O} , \mathbb {R}) and j=2,\dots , m.\end{prop}}{\begin{xmlelement*}{proof}By standard considerations there is a sequence (\phi _h)_{h\in \mathbb {N}} \subset \infty _c (\mathcal {O} , \mathbb {R}) converging uniformly to \phi  on each \mathcal {O} ^{\prime } \Subset \mathcal {O}. We would like to prove that the sequence (D_2^{\phi _h} \phi _h, \dots , D_m^{\phi _h} \phi _h)_{h\in \mathbb {N}} converges to (D^\phi _2 \phi , \dots , D^\phi _m\phi ) in the sense of distributions on each \mathcal {O} ^{\prime } \Subset \mathcal {O}. Indeed, we start to show that\begin{equation}\int _\mathcal {O} \sum _{j=2}^{m} D_j^{\phi } \phi \, \eta _j \, d \mathcal {L}^{m+n-1} =\lim _{h\rightarrow \infty } \int _\mathcal {O} \sum _{j=2}^{m} D_j ^{\phi _h} \phi _h \, \eta _j \, d \mathcal {L}^{m+n-1} \qquad \forall \, \eta \in 1_c (\mathcal {O} , \mathbb {R}^{m-1})\end{equation}We denote by \Phi _h :\mathcal {O} \rightarrow \mathbb {G} the graph map of \hat{\phi }_h=(\phi _h,0,\dots ,0) defined as \Phi _h(a):=i(a)\hat{\phi }_h(i(a)) and \mathcal {E}_h the intrinsic subgraph of \hat{\phi }_h. Therefore, by Proposition \ref {prop4.6} we know that for each \xi =(\xi _1, \dots ,\xi _m)\in 1_c (\hat{ \mathcal {O}} \cdot  \mathbb {R}^m)\begin{equation*}\int _{\mathcal {E}} \mbox{div}_\mathbb {G}\xi \, d \mathcal {L}^{m+n} = \frac{1}{C(\mathbb {W},} \int _\mathcal {O} \xi _1 \circ \Phi - \sum _{j=2}^{m} D_j^{\phi } \phi ( \xi \circ \Phi )_j \, d \mathcal {L}^{m+n-1}\end{equation*}where C(\mathbb {W},>0 is given by Lemma \ref {coro4.3}.Moreover by the uniform convergence of \phi _h to \phi  we conclude that\begin{equation*}\int _{\mathcal {E}} \mbox{div}_\mathbb {G}\xi \, d \mathcal {L}^{m+n} = \lim _{h\rightarrow \infty }\int _{\mathcal {E}_h} \mbox{div}_\mathbb {G}\xi \, d \mathcal {L}^{m+n}\end{equation*}Now we recall that 1 functions are uniformly intrinsic differentiable maps (see Theorem \ref {propC1implicauid}) and consequently they are also locally intrinsic Lipschitz maps (see Proposition \ref {lip1512DDD}). Hence we can apply Proposition \ref {prop4.6} for every \phi _h and we obtain that\begin{equation*}\int _{\mathcal {E}_h} \mbox{div} _\mathbb {G}\xi \, d \mathcal {L}^{m+n} = \frac{1}{C(\mathbb {W},} \int _\mathcal {O} \xi _1 \circ \Phi _h-\sum _{j=2}^{m} D_j^{\phi _h} \phi _h ( \xi \circ \Phi _h)_j \, d \mathcal {L}^{m+n-1}\end{equation*}for every \xi \in 1_c (\hat{ \mathcal {O}} \cdot  \mathbb {R}^m). Finally putting together the last three equalities we have that\begin{equation*}\frac{1}{C(\mathbb {W},} \int _\mathcal {O} \xi _1 \circ \Phi -\sum _{j=2}^{m} D_j^{\phi } \phi ( \xi \circ \Phi )_j \, d \mathcal {L}^{m+n-1}= \lim _{h\rightarrow \infty } \frac{1}{C(\mathbb {W},} \int _\mathcal {O} \xi _1 \circ \Phi _h-\sum _{j=2}^{m} D_j^{\phi _h} \phi _h ( \xi \circ \Phi _h)_j \, d \mathcal {L}^{m+n-1}\end{equation*}for all \xi \in 1_c (\hat{ \mathcal {O}} \cdot  \mathbb {R}^m).\end{xmlelement*}Clearly if we choose \xi _1=0, then\begin{equation}\int _\mathcal {O} \sum _{j=2}^{m} D_j^{\phi } \phi ( \xi \circ \Phi )_j \, d \mathcal {L}^{m+n-1}= \lim _{h\rightarrow \infty } \int _\mathcal {O} \sum _{j=2}^{m} D_j^{\phi _h} \phi _h ( \xi \circ \Phi _h)_j \, d \mathcal {L}^{m+n-1}.\end{equation}If we consider \xi _j(i(a)\cdot (t,0,\dots , 0)):=\eta _j(a)\rho (t) with \eta =(\eta _2, \dots , \eta _m) \in 1_c (\mathcal {O} ,\mathbb {R}^{m-1}) and \rho \in 1_c (\mathbb {R}) such that \rho (t)=1 for all t\in [-\Vert  \phi \Vert _{\mathcal {L}^\infty (\mbox{spt}(\eta ) )}-1,\Vert  \phi \Vert _{\mathcal {L}^\infty (\mbox{spt}(\eta ) )}+1], then \xi =(0, \xi _2,\dots , \xi _m) \in 1_c (\hat{ \mathcal {O}} \cdot  \mathbb {R}^m) and by (\ref {eqADHOC}) we deduce that\begin{equation*}\int _\mathcal {O} \sum _{j=2}^{m} D_j^{\phi } \phi (a) \eta _j(a)\rho (\phi (a)) \, d \mathcal {L}^{m+n-1}(a)= \lim _{h\rightarrow \infty } \int _\mathcal {O} \sum _{j=2}^{m} D_j^{\phi _h} \phi _h (a) \eta _j(a)\rho (\phi _h(a)) \, d \mathcal {L}^{m+n-1}(a).\end{equation*}Hence since \phi _h converges uniformly to \phi , there is \bar{h} \in \mathbb {N} such that for all h\ge \bar{h} and for all a\in  spt(\eta ) we have\begin{enumerate}\item \phi _h (a)\in [-M-1 ,M+1] for M:=\Vert  \phi \Vert _{\mathcal {L}^\infty (\mbox{spt}(\eta ) , \mathbb {R}),}\item \rho (\phi _h(a)) =1;\end{enumerate}and consequently (\ref {72cit}) follows.}Now using again the uniformly convergence of \phi _h to \phi  and (\ref {72cit}) we conclude that for all \eta =(\eta _2,\dots , \eta _m) \in 1_c (\mathcal {O} , \mathbb {R}^{m-1}) and for all j=2,\dots , m\begin{equation*}\begin{aligned}\int _\mathcal {O} D^\phi _j \phi \, \eta _j\, d \mathcal {L}^{m+n-1} & = \lim _{h\rightarrow \infty } \int _\mathcal {O} D^{\phi _h}_j \phi _h\, \eta _j \, d \mathcal {L}^{m+n-1} \\& = - \lim _{h\rightarrow \infty } \int _\mathcal {O} \phi _h \left( \,X_j \eta _j +\phi _h \sum _{s=1}^n b_ {j1 }^{(s)} Y_s \eta _j\right) \, d \mathcal {L}^{m+n-1}\\& = - \int _\mathcal {O} \phi \left( \,X_j \eta _j +\phi \sum _{s=1}^n b_ {j1 }^{(s)} Y_s \eta _j\right) \, d \mathcal {L}^{m+n-1}\\\end{aligned}\end{equation*}Then the thesis follows with \zeta =\eta _j \in 1_c (\mathcal {O}, \mathbb {R}) for each j=2,\dots , m.}\end{equation*}\end{equation}\)
Theorem 5.4 
Let \(\mathbb {G}:= (\mathbb {R}^{m+n}, \cdot , \delta _\lambda )\)  be a Carnot group of step 2 and \(, \) W\( the complementary subgroups defined in (\ref {5.2.0}). Let \) : O  be a locally intrinsic Lipschitz function, where \(\hat{ \mathcal {O}} \subset \mathbb {W}\)  is an open set and \(\phi :\mathcal {O} \rightarrow \mathbb {R}\)  is the map associated to \(\hat{\phi }\)  as in \((\ref {phipsi})\) .
Then there exist \(c=c(\mathbb {G})>0\)  and \(C(\mathbb {W},>0\)  such that the following area formula holds
\(|\partial \mathcal {E}|_\mathbb {G}(\hat{ \mathcal {O}} \cdot = c \, \mathcal {S}^{\mathfrak {q} -1} (\Phi (\mathcal {O} )) =\frac{1}{C(\mathbb {W},} \int _\mathcal {O} \sqrt{1+|\nabla ^{\hat{\phi }} \hat{\phi }|_{\mathbb {R}^{m-1}}^2}\, d \mathcal {L}^{m+n-1}\) 

Thanks to Proposition REF  and Theorem REF  we know that \(|\partial \mathcal {E}|_\mathbb {G}\)  is a Radon measure. Then a classical approximation result ensures the existence of sequence \((\xi _h)_{h\in \mathbb {N}}= (\xi _{h,1},\dots , \xi _{h,m})_{h\in \mathbb {N}}\subset 1_c (\hat{ \mathcal {O}} \cdot  \mathbb {R}^m)\)  with \(|\xi _h|_{\mathbb {R}^{m}}\le 1\)  such that
\(\xi _h \rightarrow \nu _\mathcal {E}\quad |\partial \mathcal {E}|_\mathbb {G}\mbox{ -a.e. in } \hat{ \mathcal {O}} \cdot {equation}and by Lemma \ref {coro4.2} and Lemma \ref {coro4.3} we have that \xi _h \circ \Phi \rightarrow \nu _\mathcal {E}\circ \Phi  a.e. in  \mathcal {O} .\) 
Moreover using () in Proposition REF  we get that for all \(h\in \mathbb {N}\)
\(- \int _{\hat{ \mathcal {O}} \cdot  \langle \xi _h , \nu _\mathcal {E}\rangle \, d|\partial \mathcal {E}|_\mathbb {G}= \frac{1}{C(\mathbb {W},} \int _\mathcal {O} \xi _{h,1 } \circ \Phi -\sum _{j=2}^{m} D_j^{\phi } \phi ( \xi _h \circ \Phi )_j \, d \mathcal {L}^{m+n-1}}Hence taking the limit as h\rightarrow \infty  in the last equality and by (\ref {convanu}) and Lemma \ref {coro4.2} we conclude\begin{equation*}|\partial \mathcal {E}|_\mathbb {G}(\hat{ \mathcal {O}} \cdot = \frac{1}{C(\mathbb {W},} \int _\mathcal {O} \sqrt{1+| \nabla ^{\hat{\phi }} \hat{\phi }|_{\mathbb {R}^{m-1}}^2}\, d \mathcal {L}^{m+n-1}.\end{equation*}Finally, |\partial \mathcal {E}|_\mathbb {G}(\hat{ \mathcal {O}} \cdot = c \, \mathcal {S}^{\mathfrak {q} -1} (\Phi (\mathcal {O} )) is a direct consequence of results of Theorem \ref {Theorem 4.18fssc} and Theorem \ref {teo433}.\) 
Smooth approximation for intrinsic lipschitz maps
In this section we characterize intrinsic Lipschitz functions as maps which can be approximated by a sequence of smooth maps, with pointwise convergent intrinsic gradient.
The classical approximation by convolution does not apply because of the nonlinearity of \(\nabla ^{\hat{\phi }} \hat{\phi }\) . Here we use the similar technique in [39]} in the context of Heisenberg groups. The strategy is somehow indirect, indeed we approximate in Carnot groups of step 2 the intrinsic subgraph of intrinsic Lipschitz function \(\hat{\phi }\) , rather than the function itself. The key point is that the intrinsic subgraph of \(\hat{\phi }\)  is a set of locally finite \(\mathbb {G}\) -perimeter (see Proposition REF ). Precisely, we prove the following Theorem
Theorem 6.1 
Let \(\mathbb {G}:= (\mathbb {R}^{m+n}, \cdot , \delta _\lambda )\)  be a Carnot group of step 2 and \(, \) W\( the complementary subgroups defined in (\ref {5.2.0}). Let \) : O  be a locally intrinsic Lipschitz function, where \(\hat{ \mathcal {O}}\)  is an open subset of \(\mathbb {W}\)  and \(\phi :\mathcal {O} \rightarrow \mathbb {R}\)  is the map associated to \(\hat{\phi }\)  as in \((\ref {phipsi})\) .
Then if \(\nabla ^{\hat{\phi }} \hat{\phi }\in \mathcal {L}^\infty _{loc}(\hat{ \mathcal {O}} ,\mathbb {R}^{m-1})\)  there exists a sequence \((\hat{\phi }_h)_h \subset \infty (\hat{ \mathcal {O}} ,\)  such that for all \(\hat{ \mathcal {O}^{\prime }} \Subset \hat{ \mathcal {O}}\)  there is \(C=C(\mathcal {O}^{\prime })>0\)  such that

\(\hat{\phi }_h\)  uniformly converges to \(\hat{\phi }\)  in \(\hat{ \mathcal {O}^{\prime }} \)

\(|\nabla ^{\hat{\phi }_h} \hat{\phi }_h(a) | \le C\)  \(\forall a\in \hat{ \mathcal {O}^{\prime }} \) , \(h\in \mathbb {N}\)

\(\nabla ^{\hat{\phi }_h} \hat{\phi }_h (a)\rightarrow \nabla ^{\hat{\phi }} \hat{\phi }(a)\)  a.e. \(a\in \hat{ \mathcal {O}^{\prime }}\)


Before stating the approximation theorem we need to recall the following result.
Theorem 6.2 ([40]}) 
Let \(f:\mathbb {R}^{m+n} \rightarrow \mathbb {R}\)  be a strictly convex function and let \((g_h)_h\)  and \(g\)  be in \(\mathcal {L}^1(\Omega , \mathbb {R}^{m+n})\)  where \(\Omega \subset \mathbb {R}^{m+n}\) . If

\(g_h \rightarrow g \, \, \)  weakly in \(\mathcal {L}^1(\Omega , \mathbb {R}^{m+n})\)

\(\int _\Omega f \circ g_h \)  d\( \mathcal {L}^{m+n} \rightarrow \int _\Omega f \circ g \)  d\( \mathcal {L}^{m+n}\)

then \(g_h \rightarrow g \,\)  strongly in \(\mathcal {L}^1(\Omega , \mathbb {R}^{m+n})\) .

Let \(\mathcal {O}^{\prime } \Subset \mathcal {O}\)  and \(M:=\Vert  \phi \Vert _{\mathcal {L}^\infty (\mathcal {O}^{\prime }, \mathbb {R})} <+\infty \) .
We split the proof in several steps. First we use Friedrichs' mollifier to regularize \(\chi _{E }\)  with a family of functions \((f_\alpha )_{\alpha >0}\)  which satisfies Theorem REF . Then we consider the level set of \((f_\alpha )_{\alpha >0}\)  which defines a family of functions \((\phi _\alpha )_{\alpha >0}\) . Finally we prove that \((\phi _\alpha )_{\alpha >0}\)  is the family of functions, up to subsequence, that we were looking for.
\(\mathbf {Step \, 1.}\)  Let \(\mathcal {E}:=\mathcal {E}_{\hat{\phi }}\)  the intrinsic subgraph of \(\hat{\phi }|_{\mathcal {O}^{\prime } }\)  defined as (). For every \(\alpha >0\)  we consider \(f_\alpha :\mathbb {G}\rightarrow \mathbb {R}\)  given by
\(\begin{aligned}f_\alpha (p):= (\rho _\alpha \ast \chi _{\mathcal {E}})(p)& =\int _\mathbb {G}\rho _\alpha (pq^{-1}) \chi _{\mathcal {E}}(q)\, d \mathcal {L}^{m+n}(q)\\& =\int _\mathbb {G}\rho _\alpha (q) \chi _{\mathcal {E}}(q^{-1} p)\, d \mathcal {L}^{m+n}(q)\end{aligned}\) 
where \(\rho _\alpha (p):=\alpha ^{m+2n}\rho (\delta _{1/\alpha }(p))\)  and \(\rho \in \infty _c (\mathcal {U}(0,1), \mathbb {R})\)  is a smooth mollifier with \(\rho (p^{-1})=\rho (-p)=\rho (p)\)  for every \(p\in \mathbb {G}\) .
By properties of convolution in \(\mathbb {G}\)  introduced in [41]}, we know that \(f_\alpha \in \infty _c (\mathbb {G}, \mathbb {R})\)  and spt\((f_\alpha )\subset \mathcal {U}(0,\alpha )\cdot \)  spt\((\chi _{\mathcal {E}})\)  for every \(\alpha >0\) . Precisely, for all \(\alpha >0\)
\(f_\alpha (p)\in [0,1] \quad \mbox{ for all } p\in \mathbb {G}\) 
and for all sufficiently small \(\alpha >0\)
\(f_\alpha (p)=1 \quad \mbox{ for all } p=(p_1,\dots , p_{m+n})\in \mathbb {G}\, \mbox{ such that } p_1 \le -2M\) 
Moreover by definition of \(\mathcal {E}\)  we know that \(\mathcal {E}\)  is an open set in \(\mathbb {G}\)  and
\(\mbox{ spt} (\chi _{\mathcal {E}}) = \mbox{clos} ( \mathcal {E}) \subset \lbrace a\cdot (t,0,\dots , 0 )\in \mbox{clos} (\hat{ \mathcal {O}^{\prime }} ) \cdot :\, t \le M \rbrace .\) 
As a consequence
\(\mbox{ spt} (f_\alpha ) \subset \overline{\mathcal {U}(0,\alpha )}\cdot \mbox{ spt} (\chi _{\mathcal {E}})\subset \lbrace a\cdot (t,0,\dots , 0 )\in \hat{ \mathcal {O}^{\prime }} \cdot :\, t < 2M \rbrace  \quad \mbox{ for } \alpha <M\) 
and
\(f_\alpha (p)=0 \quad \mbox{ for all } p=(p_1,\dots , p_{m+n})\in \mathbb {G}\, \mbox{ such that } p_1 \ge 2M\) 
i.e. \(f_\alpha \)  is constant far from the graph of \(\hat{\phi }\) , so \((\ref {conv})\)  is extended only in a neighborhood of the graphs itself.
\(\mathbf {Step \, 2.}\)  For every \(\alpha >0\)  and \(c\in (0,1)\) , let
\(S_\alpha := \left\lbrace  p=(p_1,\dots , p_{m+n})\in \hat{ \mathcal {O}^{\prime }} \cdot  : \, p_1 \in (-2M,2M),\, f_\alpha (p)=c \right\rbrace .\) 
We note that if
\(\mbox{rank} \nabla _\mathbb {G}f_\alpha (p)=1, \quad \mbox{for all } p\in S_\alpha \) 
then \(S_\alpha \)  is the level set of a map \(f_\alpha \in \infty (\mathbb {G}, \mathbb {R})\)  and so \(f_\alpha \in 1_\mathbb {G}(\mathbb {G}, \mathbb {R})\)  such that \(\mbox{rank} \nabla _\mathbb {G}f_\alpha \)  is maximum, i.e. \(S_\alpha \)  is \(\mathbb {G}\) -regular hypersurface. More precisely, we show that
\(X_1f_\alpha (p) <0 \quad \mbox{for all } p\in S_\alpha \) 
where \(X_1f_\alpha (p)\)  is the first component of horizontal gradient \(\nabla _\mathbb {G}f_\alpha \)  of \( f_\alpha \)  in \(p\) . Indeed let \(\mathcal {A} ^{\prime }:=\lbrace p= (p_1,\dots , p_{m+n})\in \hat{ \mathcal {O}^{\prime }} \cdot  :\, p_1\in (-3M,3M) \rbrace \)  and \(\varphi \in \infty _c(\mathcal {A} ^{\prime } , \mathbb {R})\) , then
\(\begin{aligned}\langle X_1f_\alpha , \varphi \rangle &=-\int _{ \mathcal {A} ^{\prime } } f_\alpha (q)X_1 \varphi (q) \, d\mathcal {L}^{m+n} (q)\\&=-\int _{ \overline{\mathcal {U}(0,\alpha )} } \rho _\alpha (p) \, d\mathcal {L}^{m+n} (p) \int _{ \mathcal {A} ^{\prime } } \chi _\mathcal {E}(p^{-1}q) X_1 \varphi (q)\, d\mathcal {L}^{m+n} (q)\\&=-\int _{ \overline{\mathcal {U}(0,\alpha )} } \rho _\alpha (p) \, d\mathcal {L}^{m+n} (p) \int _{\tau _{P^{-1}} (\mathcal {A} ^{\prime }) } \chi _\mathcal {E}(q) X_1 \varphi (pq)\, d\mathcal {L}^{m+n} (q)\\\end{aligned}\) 
With the notation \(\varphi _p(q):=\varphi (pq)\)  we have \(X_1(\varphi (pq))= X_1 \varphi _p (q)\)  because \(X_1\)  is left-invariant; moreover \(\varphi _p \in \infty _c (\tau _{p^{-1}} \mathcal {A} ^{\prime } , \mathbb {R})\)  and by integration by parts, we obtain
\(\begin{aligned}\int _{\tau _{P^{-1}} (\mathcal {A} ^{\prime }) } \chi _\mathcal {E}(q) X_1 \varphi _P(q)\, d\mathcal {L}^{m+n} (q) = - \int _{\tau _{p^{-1}} (\mathcal {A} ^{\prime }) } \nu ^{(1)}_{\mathcal {E}} (q) \varphi _P(q)\, d |\partial \mathcal {E}|_\mathbb {G}(q)\end{aligned}\) 
Since \(\mbox{spt}(\varphi _P) \Subset \tau _{P^{-1}} (\mathcal {A} ^{\prime })\)  and \(P\in \overline{\mathcal {U}(0,\alpha )}\)  if \(\alpha \)  is small enough, we can replace \(\tau _{p^{-1}} (\mathcal {A} ^{\prime })\)  by \(\mathcal {A} ^{\prime }\) . Thus by Fubini-Tonelli Theorem and a change of variable, we get
\(\begin{aligned}\langle X_1f_\alpha , \varphi \rangle &=\int _{\mathcal {A} ^{\prime } } \nu ^{(1)}_{\mathcal {E}} (q) \, d |\partial \mathcal {E}|_\mathbb {G}(q) \left( \int _{\mathbb {G}} \rho _\alpha (p) \varphi (pq) \, d\mathcal {L}^{m+n} (p) \right)\\\end{aligned}\) 
Then for all \(p\in \mathcal {A} ^{\prime }\)  and for all \(\alpha >0\)  small enough
\(\begin{aligned}X_1f_\alpha (p) & = \int _{ \mathcal {A} ^{\prime } }\rho _\alpha (p q^{-1}) \nu ^{(1)}_{\mathcal {E}} (q)\, d |\partial \mathcal {E}|_\mathbb {G}(q)\\&= \int _{\mathcal {U}(0,\alpha )\cdot P }\rho _\alpha (p q^{-1}) \nu ^{(1)}_{\mathcal {E}} (q)\, d |\partial \mathcal {E}|_\mathbb {G}(q)\end{aligned}\) 
In particular we immediately deduce from \((\ref {83cit})\)  the following assertion: For all couple \((\hat{ \mathcal {O}^{\prime }} ,\hat{ \mathcal {O}} _0^{\prime })\)  of open and bounded subsets of \(\mathbb {W}\)  with \(\hat{ \mathcal {O}^{\prime }} \Subset \hat{ \mathcal {O}} _0^{\prime }\)  there is \(\bar{\alpha }=\alpha (\hat{ \mathcal {O}} _0^{\prime })>0\)  such that for every \(0<\alpha <\bar{\alpha }\)
\(\int _{ \mathcal {A} }|\nabla _\mathbb {G}f_\alpha |\, d \mathcal {L}^{m+n} \le |\partial \mathcal {E}|_\mathbb {G}( \mathcal {A}_0 ).\) 
where \( \mathcal {A}:=\lbrace p= (p_1,\dots , p_{m+n})\in \hat{ \mathcal {O}^{\prime }} \cdot  :\, p_1\in (-2M,2M) \rbrace \)  and \( \mathcal {A}_0 :=\lbrace p= (p_1,\dots , p_{m+n})\in \hat{ \mathcal {O}}_0^{\prime } \cdot  :\, p_1\in (-2M,2M) \rbrace \) . Moreover since Lemma \(\ref {coro4.2}\)  we get
\(\nu ^{(1)}_{\mathcal {E}} \circ \Phi =-\frac{1}{\sqrt{1+|\nabla ^{\hat{\phi }} \hat{\phi }|_{ \mathbb {R}^{m-1}}^2}}\quad \mbox{in } \mathcal {O}^{\prime }\) 
and if we put
\(I_\alpha (p):= \int _{ \mathcal {U}(0,\alpha )\cdot p}\rho _\alpha (pq^{-1})\, d |\partial \mathcal {E}|_\mathbb {G}(q)\) 
then using \((\ref {83cit})\)  we have
\(X_1f_\alpha (p) \le -\frac{1}{\sqrt{1+\Vert \nabla ^{\hat{\phi }} \hat{\phi }\Vert ^2_{\mathcal {L}^\infty (\hat{ \mathcal {O}^{\prime }} , \mathbb {R}^{m-1})}}}I_\alpha (p) \quad \mbox{ for all } p \in \mathcal {A}^{\prime } .\) 
Moreover for every fixed \(c\in (0,1)\)  and for all \(p\in \mathcal {A}\)  such that \(f_\alpha (p)=c\) , then
\(\mathcal {L}^{m+n}((\mathcal {U}(0,\alpha )\cdot p) \cap \mathcal {E})>0 \quad \mbox{and} \quad \mathcal {L}^{m+n}((\mathcal {U}(0,\alpha )\cdot p) \cap \mathcal {E}^c )>0.\) 
Indeed, by contradiction, if we assume \( \mathcal {L}^{m+n}((\mathcal {U}(0,\alpha )\cdot p) \cap \mathcal {E})=0\) . Then because \(\mathcal {E}\)  is open, we can assume \((\mathcal {U}(0,\alpha )\cdot p) \cap \mathcal {E}= \emptyset \) . Hence by the definition of convolution, we have that \(f_\alpha (p)=0 \ne c\)  and then a contradiction. In the similar way it follows that \(f_\alpha (p)=1\)  if we suppose by contradiction \( \mathcal {L}^{m+n}((\mathcal {U}(0,\alpha )\cdot p) \cap \mathcal {E}^c)=0\) .
Now by \((\ref {cit88})\)  and by Theorem REF , we have that \(|\partial \mathcal {E}|_\mathbb {G}((\mathcal {U}(0,\alpha )\cdot p )>0\)  and \(I_\alpha (p )>0 \)  for \(p\in \mathcal {A} \)  with \(f_\alpha (p)=c\) . As a consequence using \((\ref {cit87})\) , \((\ref {cit85})\)  holds and \(S_\alpha \)  is a \(\mathbb {G}\) -regular hypersurface.
\(\mathbf {Step \, 3.}\)  Now we show that \((S_\alpha )_\alpha \)  implicitly defines a sequence \((\hat{\phi }_\alpha )_{\alpha }\)  with \(\hat{\phi }_\alpha =(\phi _\alpha ,0,\dots , 0)\)  and ,up to subsequence, it is the family of functions which we were looking for.
Fix \(\alpha \)  and \(c\in (0,1)\) . Because \(f_\alpha \)  is continuous map such that
\(\begin{aligned}f_\alpha ( i(a) \cdot (-2M,0,\dots , 0)) & =1>c \\f_\alpha (i(a)\cdot (2M,0,\dots , 0)) & = 0<c\end{aligned}\) 
for all \(a\in \mathcal {O}^{\prime } \) , there is \(t_a\in (-2M,2M)\)  such that \(f_\alpha ( i(a) \cdot (t_a,0 , \dots , 0))=c\) . In particular since
\(X_1f_\alpha ( i(a) \cdot (t,0,\dots , 0) )\le 0 \quad \forall \, t<t_a \, \qquad \mbox{ and } \qquad X_1f_\alpha ( i(a) \cdot (t_a,0,\dots , 0)) < 0\) 
we have that \(t_a\)  is unique and \( \lbrace  t\in (-2M,2M) \, :\, f_\alpha ( i(a) \cdot (t,0,\dots , 0))>c \rbrace =(-2M,t_a)\) .
In the similar way in Lemma 4.8 in [42]}, we define \(\hat{\phi }_\alpha (i(a)):=(\phi _\alpha (a),0,\dots , 0)\)  with \(\phi _\alpha (a):=t_a\)  and
\(\mathcal {E}_\alpha =\mathcal {E}_{\alpha , c}:= \lbrace  (i(a) \cdot (t,0,\dots , 0)) \in \hat{ \mathcal {O}} \cdot  :\, f_\alpha (i(a)\cdot (t,0,\dots , 0))>c\rbrace .\) 
Then \(\phi _\alpha :\mathcal {O}^{\prime } \rightarrow [-2M, 2M]\)  and
\(\mathcal {E}_\alpha \cap \mathcal {A}^{\prime \prime } =\mathcal {E}_{\hat{\phi }_\alpha } \cap \mathcal {A} ^{\prime \prime },\) 
where \( \mathcal {A}^{\prime \prime } :=\lbrace p\in \hat{ \mathcal {O}^{\prime }} \cdot  :\, p_1\in [-2M, 2M] \rbrace \) . Moreover recalling that \(c\in (0,1)\) , from \((\ref {cit76})\)  and \((\ref {cit79})\)  we get
\(S_\alpha = \mathrm {graph}\,(\hat{\phi }_\alpha ) =\partial \mathcal {E}_\alpha \cap ( \hat{ \mathcal {O}^{\prime }} \cdot \) 
and by Theorem REF  \(\hat{\phi }_\alpha \)  is uniformly intrinsic differentiable in \(\hat{ \mathcal {O}^{\prime }} \) .
Let \(\tilde{f} _\alpha :\mathbb {R}^{m+n} \rightarrow \mathbb {R}\)  defined by
\( \tilde{f}_\alpha (t,a ):=f_\alpha (i(a)\cdot (t,0,\dots , 0)).\) 
By \(f_\alpha \in \infty (\mathbb {G}, \mathbb {R})\)  and (REF ), \( \tilde{f}_\alpha \)  is \(\infty \)  map such that \(\frac{ \partial \tilde{f}_\alpha }{\partial p_1}(p) = X_1 f_\alpha (p) \ne 0\)  for all \(p\in S_\alpha \) . Hence we apply the classical Implicit Function Theorem to \(\tilde{f} _\alpha \)  and we conclude that also \(\hat{\phi }_\alpha \)  is \(\infty \) .
\(\mathbf {Step \, 4.}\)  Firstly we prove that \((\hat{\phi }_\alpha )_\alpha \) , up to subsequence, converges uniformly on \(\hat{ \mathcal {O}^{\prime }}\) , i.e. the condition (1) of the thesis follows.
In particular, we would like to show that there is a constant \(L>0\)  dependent on \( \Vert \nabla ^{\hat{\phi }} \hat{\phi }\Vert _{\mathcal {L}^\infty (\hat{ \mathcal {O}^{\prime }} , \mathbb {R}^{m-1})} \) , \(\mathcal {B}^{(1)}, \dots ,\mathcal {B}^{(n)}\)  such that for all \(\alpha >0\)  sufficiently small and all \(A\in \mathcal {O}^{\prime }\) , it holds
\(|\phi _\alpha (a)-\phi (a)|\le L\alpha .\) 
Therefore \((\phi _\alpha )_\alpha \) , up to subsequence, converges uniformly on \(\mathcal {O}^{\prime }\) . This is a direct consequence of Theorem REF . Here \(\nu ^\perp (p) =\mathbf {P}_\mathbb {W}(p)\) , \(\nu (p) =\mathbf {P}_p)\)  and \(E:=\mathcal {E}_{\hat{\phi }_\alpha } \)  is the intrinsic subgraph of \(\hat{\phi }_\alpha \) . Because \(\hat{\phi }_\alpha \)  is locally intrinsic Lipschitz, \(\mathcal {E}_{\hat{\phi }_\alpha } \)  is also a finite \(\mathbb {G}\) -perimeter set (see Theorem REF ).
For \(\beta \)  fixed as in Theorem REF  there is \(L=L(\beta , \mathcal {B}_M)>0\) , where \(\mathcal {B}_{M} = \max \lbrace  b_{ij}^{(s)} \, | \, i,j=1,\dots , m \, , s=1,\dots , n \rbrace \)  such that
\(\mathcal {U}((-L,0,\dots ,0),1) \subset \lbrace  q\, :\, \Vert  \mathbf {P}_\mathbb {W}q\Vert <-\beta \mathbf {P}_q̭ \rbrace  \quad \mbox{and} \quad \mathcal {U}((L,0,\dots ,0),1) \subset \lbrace  q\, :\, \Vert  \mathbf {P}_\mathbb {W}q\Vert <\beta \mathbf {P}_q̭ \rbrace \) 
Let \(p_a:= i(a) \cdot \hat{\phi }(i(a))\) , \(p^{\prime }_a:= i(a) \cdot (\phi (a)-\alpha L ,0,\dots , 0)\)  and \(p^{\prime \prime }_a:= i(a) \cdot (\phi (a)+\alpha L ,0,\dots , 0)\)  with \(a \in \mathcal {O}^{\prime } \)  and \(\alpha \in (0,1]\) . Notice that if we put \(r_0:= M+L+\max _{(x,y)\in \mathcal {O}^{\prime } }|x|_{\mathbb {R}^{m-1}}+ \max _{(x,y)\in \mathcal {O}^{\prime } }|y|_{\mathbb {R}^n}^{1/2} +\frac{1}{2} \mathcal {B}_M^{1/2} \max _{(x,y)\in \mathcal {O}^{\prime } }|x|_{\mathbb {R}^{m-1}}^{1/2} (M+L)^{1/2}< +\infty \)  then
\(\Vert  p^{\prime }_a\Vert  \le r_0,\quad \Vert  p^{\prime \prime }_a\Vert  \le r_0,\) 
for all \(a\in \mathcal {O}^{\prime }\) . Moreover by standard considerations (see [43]}) we know that for every \(q\in \mathcal {U}(0,r_0)\)  there exists \(c(r_0)>0\)  such that
\(\mathcal {U}(0,r)\cdot q \subset \mathcal {U}(q,c(r_0)\sqrt{r} ), \quad \forall \, r \in (0,1).\) 
So if we consider \(a\in \mathcal {O}^{\prime }\)  and \(\alpha \in (0,1]\) , we conclude that
\(\begin{aligned}\mathcal {U}(0,\alpha )\cdot p^{\prime }_a & \subset \mathcal {U}(p^{\prime }_a,c(r_0)\sqrt{\alpha }) \subset \lbrace  q\, :\, \Vert  \mathbf {P}_\mathbb {W}(p_a ^{-1} Q)\Vert <-\beta \Vert \mathbf {P}_p_a ^{-1} q) \Vert  \rbrace  \subset \mathcal {E}_{\hat{\phi }_\alpha }\\\mathcal {U}(0,\alpha )\cdot p^{\prime \prime }_a& \subset \mathcal {U}(p^{\prime \prime }_a,c(r_0)\sqrt{\alpha }) \subset \lbrace  q\, :\, \Vert  \mathbf {P}_\mathbb {W}(p_a ^{-1} Q)\Vert <\beta \Vert \mathbf {P}_p_a^{-1} q)\Vert  \rbrace  \subset \mathbb {G}- \mathcal {E}_{\hat{\phi }_\alpha }\end{aligned}\) 
More precisely, by definition of \(f_\alpha \)  we have
\(f_\alpha (p^{\prime }_a)=1, \quad \quad f_\alpha \left( i(a) \cdot \hat{\phi }_\alpha (i(a)) \right)=c, \quad \quad f_\alpha (p^{\prime \prime }_a)=0\) 
and so by \((\ref {90cit})\) , \((\ref {91cit})\)  and \((\ref {cit85})\)  we deduce
\(\phi (a)-\alpha L \le \phi _\alpha (a) \le \phi (a)+\alpha L\quad \mbox{ for all } \alpha \in (0,1], \, a\in \mathcal {O}^{\prime }.\) 
Hence (REF ) holds and consequently the condition (1) of the thesis is true.
\(\mathbf {Step \, 5.}\)  We prove that \((\hat{\phi }_\alpha )_{\alpha }\)  has a subsequence \((\hat{\phi }_k)_k\)  such that \(|\nabla ^{\hat{\phi }_k} \hat{\phi }_k |_{\mathbb {R}^{m-1}} \le \Vert  \nabla ^{\hat{\phi }} \hat{\phi }\Vert _{\mathcal {L}^\infty (\hat{ \mathcal {O}^{\prime }} ,\mathbb {R}^{m-1})}\)  for each \(k\in \mathbb {N}\)  on \(\mathcal {O} ^{\prime }\) , i.e. the condition \((2)\)  of the thesis holds.
Let \(\hat{\nabla }_\mathbb {G}f_\alpha :=(X_2f_\alpha , \dots ,X_mf_\alpha )\)  and \(\hat{\nu }_{\mathcal {E}} =(\nu ^{(2)}_{\mathcal {E}},\dots , \nu ^{(m)}_{\mathcal {E}})\) . Arguing as in Step 2 and by (), it follows
\(\begin{aligned}| \nabla ^{\hat{\phi }_\alpha } \hat{\phi }_\alpha (p)|_{\mathbb {R}^{m-1}} &= \frac{|\hat{\nabla }_\mathbb {G}f_\alpha (p)|_{\mathbb {R}^{m-1}}}{|X_1f_\alpha (p)|} \le \frac{1}{|X_1f_\alpha (p)|} \int _{ \mathcal {U}(0,\alpha )\cdot p} |\hat{\nu }_{\mathcal {E}} (q)| |\rho _\alpha (p\cdot q^{-1})| \, d |\partial \mathcal {E}|(q)\\& \le I_\alpha (p) \frac{ \Vert  \nabla ^{\hat{\phi }} \hat{\phi }\Vert _{\mathcal {L}^\infty (\hat{ \mathcal {O}^{\prime }},\mathbb {R}^{m-1})} }{|X_1f_\alpha (p)| \sqrt{1+ \Vert  \nabla ^{\hat{\phi }} \hat{\phi }\Vert ^2_{\mathcal {L}^\infty (\hat{ \mathcal {O}^{\prime }} ,\mathbb {R}^{m-1})} }} \\\end{aligned}\) 
and consequently by \((\ref {cit87})\)
\(| \nabla ^{\hat{\phi }_\alpha } \hat{\phi }_\alpha (p)|_{\mathbb {R}^{m-1}} \le \Vert  \nabla ^{\hat{\phi }} \hat{\phi }\Vert _{ \mathcal {L}^\infty (\hat{ \mathcal {O}^{\prime }} ,\mathbb {R}^{m-1})},\) 
for all \(p\in \hat{ \mathcal {O}^{\prime }}\) , as desired.
\(\mathbf {Step \, 6.}\)  Now we show that \((\hat{\phi }_\alpha )_\alpha \) , up to a subsequence, satisfying the condition \((3)\)  of the thesis, i.e. there exists a sequence \((\alpha _h)_h \subset (0,+\infty )\)  such that let \(\hat{\phi }_h \equiv \hat{\phi }_{\alpha _h}\)  we have
\(\nabla ^{\hat{\phi }_h} \hat{\phi }_h \rightarrow \nabla ^{\hat{\phi }} \hat{\phi }\quad \mbox{ a.e. in } \hat{ \mathcal {O}^{\prime }}.\) 
It is sufficient to show that there is a positive sequence \((\alpha _h)_h\)  converging to 0 such that there exists
\(\lim _{h\rightarrow \infty } \int _{ \mathcal {O}^{\prime }} \sqrt{ 1+|\nabla ^{\hat{\phi }_h} \hat{\phi }_h|_{\mathbb {R}^{m-1}} ^2} \, d \mathcal {L}^{m+n-1} = \int _{ \mathcal {O}^{\prime }} \sqrt{ 1+|\nabla ^{\hat{\phi }} \hat{\phi }|_{\mathbb {R}^{m-1}} ^2} \, d \mathcal {L}^{m+n-1}\) 
In fact, up a subsequence, using \((\ref {cit93})\)  and Proposition \(\ref {theo4.7}\)  we can assume that the sequence in \((\ref {citt98})\)  also satisfies
\(\nabla ^{\hat{\phi }_h} \hat{\phi }_h \rightarrow \nabla ^{\hat{\phi }} \hat{\phi }\quad \mbox{ weakly in }\, \mathcal {L}^1( \hat{ \mathcal {O}^{\prime }}, \mathbb {R}^{m-1} ).\) 
Consequently by Theorem  REF
\(\nabla ^{\hat{\phi }_h} \hat{\phi }_h \rightarrow \nabla ^{\hat{\phi }} \hat{\phi }\quad \mbox{ strongly in }\, \mathcal {L}^1( \hat{ \mathcal {O}^{\prime }}, \mathbb {R}^{m-1} )\) 
and so, up a subsequence, \((\ref {citt97})\)  holds.
In particular in order to show \((\ref {citt98})\)  we need to prove that there exist \(\hat{c} \in (0,1)\)  and a positive sequence \((\alpha _h)_h\)  converging to 0 satisfying
\(\exists \, \lim _{h\rightarrow \infty } |\partial \mathcal {E}_{\alpha _h, \hat{c} }|_\mathbb {G}( \mathcal {A}) = |\partial \mathcal {E}|_\mathbb {G}(\mathcal {A})\) 
where \(\mathcal {A}\)  is defined as in Step 2. Since the semicontinuity of \(\mathbb {G}\) -perimeter measure and Step 3 we obtain
\(|\partial \mathcal {E}|_\mathbb {G}(\mathcal {A} ) \le \liminf _{\alpha \rightarrow 0^+} |\partial \mathcal {E}_{\alpha , c }|_\mathbb {G}(\mathcal {A} )\) 
for every \(c\in (0,1)\) . More precisely by the last inequality and the coarea formula we get
\(\begin{aligned}|\partial \mathcal {E}|_\mathbb {G}(\mathcal {A}) & \le \int _0^1 \liminf _{\alpha \rightarrow 0^+} |\partial \mathcal {E}_{\alpha , c }|_\mathbb {G}(\mathcal {A} )\, dc \le \liminf _{\alpha \rightarrow 0^+} \int _0^1 |\partial \mathcal {E}_{\alpha , c }|_\mathbb {G}(\mathcal {A} )\, dc \\ & \quad = \liminf _{\alpha \rightarrow 0^+} \int _{\mathcal {A} } | \nabla _\mathbb {G}f_\alpha |\, d \mathcal {L}^{m+n} =: I(\hat{ \mathcal {O}^{\prime }} , c ).\end{aligned}\) 
Now using Step 2 we know that for every \(\hat{ \mathcal {O}}_0^{\prime }\)  such that \(\hat{ \mathcal {O}^{\prime }}\Subset \hat{ \mathcal {O}} _0^{\prime } \)  open and bounded there exists \((\alpha _h)_h \subset (0,+ \infty )\)  which converges to 0 and \( \bar{h} (\hat{ \mathcal {O}}_0^{\prime })>0\)  such that for all \(h\le \bar{h} (\hat{ \mathcal {O}}_0^{\prime })\)  we have
\(\int _{\mathcal {A} } | \nabla _\mathbb {G}f_{\alpha _h}| \, d \mathcal {L}^{m+n} \le |\partial \mathcal {E}|_\mathbb {G}( \mathcal {A}_0)\) 
where \(\mathcal {A}, \mathcal {A}_0\)  are defined as in Step 2. Consequently
\(I(\hat{ \mathcal {O}^{\prime }}, c ) \le |\partial \mathcal {E}|_\mathbb {G}(\mathcal {A}_0)\) 
for all \(c\in (0,1)\)  and \(\hat{ \mathcal {O}}_0^{\prime }\)  such that \(\hat{ \mathcal {O}^{\prime }} \Subset \hat{ \mathcal {O}}_0^{\prime } \)  open and bounded. Moreover by \(|\partial \mathcal {E}|_\mathbb {G}\)  is a Radon measure then by standard approximation argument we can rewrite \((\ref {cit109})\)  with \(\hat{ \mathcal {O}^{\prime }}\)  instead of \(\hat{ \mathcal {O}}_0^{\prime }\) . In particular using \((\ref {cit106})\)  we have that a.e. \(c\in (0,1)\)
\(\liminf _{\alpha \rightarrow 0^+} |\partial \mathcal {E}_{\alpha , c }|_\mathbb {G}(\mathcal {A} )= |\partial \mathcal {E}|_\mathbb {G}(\mathcal {A} ).\) 
Hence there exists \(\hat{c} \in (0,1)\)  and a positive sequence \((\alpha _h)_h\)  converging to 0 such that \((\ref {cit101})\)  follows.
Finally because \(\Vert  \phi \Vert _{\mathcal {L}^\infty (\mathcal {O} ^{\prime } , \mathbb {R}) } = M\)  we get
\(\partial \mathcal {E}\cap \mathcal {A} _1 = \partial \mathcal {E}\cap \mathcal {A} _2 = \emptyset \) 
where \( \mathcal {A}_1:=\lbrace p \in \hat{ \mathcal {O}^{\prime }} \cdot  :\, p_1 \le -2M \rbrace \)  and \(\mathcal {A}_2:=\lbrace p\in \hat{ \mathcal {O}^{\prime }} \cdot  :\, p_1 \ge 2M \rbrace \) .
Hence since Proposition \(\ref {citteo16OK}\)  and well-know \(\mathbb {G}\) -perimeter properties it follows that
\(\begin{aligned}\frac{1}{C(\mathbb {W},} \int _{ \mathcal {O}^{\prime }} \sqrt{1+| \nabla ^{\hat{\phi }} \hat{\phi }|_{\mathbb {R}^{m-1}} ^2} \, d\mathcal {L}^{m+n-1} & =|\partial \mathcal {E}|_\mathbb {G}( \hat{ \mathcal {O}^{\prime }} \cdot  \\& = |\partial \mathcal {E}|_\mathbb {G}(\mathcal {A} _1 )+ |\partial \mathcal {E}|_\mathbb {G}(\mathcal {A} )+ |\partial \mathcal {E}| ( \mathcal {A}_2 )\\& = |\partial \mathcal {E}|_\mathbb {G}( \partial \mathcal {E}\cap \mathcal {A}_1) + |\partial \mathcal {E}|_\mathbb {G}(\mathcal {A} ) + |\partial \mathcal {E}|_\mathbb {G}( \partial \mathcal {E}\cap \mathcal {A}_2 )\\&= |\partial \mathcal {E}|_\mathbb {G}( \mathcal {A} ).\end{aligned}\) 
In the same way, using also \((\ref {cit76})\) , \((\ref {cit79})\)  and \((\ref {90cit})\)
\(|\partial \mathcal {E}_{\alpha _h , c} |_\mathbb {G}( \mathcal {A} )= |\partial \mathcal {E}_{\hat{\phi }_h }|_\mathbb {G}( \mathcal {A} )\) 
and
\(|\partial \mathcal {E}_{\hat{\phi }_h }|_\mathbb {G}( \hat{ \mathcal {O}^{\prime }} \cdot = \frac{1}{C(\mathbb {W},} \int _{ \mathcal {O}^{\prime }} \sqrt{1+|\nabla ^{\hat{\phi }_h} \hat{\phi }_h |_{\mathbb {R}^{m-1}} ^2} \, \, d \mathcal {L}^{m+n-1}\) 
with \(\hat{\phi }_h = \hat{\phi }_{\alpha _h }\) . Now by the last three equalities and \((\ref {cit101})\) , \((\ref {citt98})\)  holds. Consequently \((\ref {citt97})\)  is true and the proof of theorem is complete.

Intrinsic Lipschitz Function and Distributional Solution
The main results in this section are Theorem REF  and Theorem REF . We show that a map \(\hat{\phi }=(\phi , 0 \dots , 0): \hat{ \mathcal {O}} \rightarrow  is a locally intrinsic Lipschitz function if and only if \)\( is locally \) 1/2 \(-Hölder continuous and it is also distributional solution of system \)\(\left(D^{\phi }_2\phi ,\dots , D^{\phi }_m\phi \right) =w \quad \mbox{in } \mathcal {O}.\)\(More precisely, we generalize Theorem 1.1 on \cite {biblio17} valid in Heisenberg groups.\)
Theorem 7.1 
Let \(\mathbb {G}:= (\mathbb {R}^{m+n}, \cdot , \delta _\lambda )\)  be a Carnot group of step 2 and \(, \) W\( the complementary subgroups defined in (\ref {5.2.0}). Let \) : O  be a locally intrinsic Lipschitz function, where \(\hat{ \mathcal {O}}\)  is an open subset of \(\mathbb {W}\)  and \(\phi :\mathcal {O} \rightarrow \mathbb {R}\)  is the map associated to \(\hat{\phi }\)  as in \((\ref {phipsi})\) .
Then if \((D^{\phi }_2\phi ,\dots , D^{\phi }_m\phi ) \in \mathcal {L}_{loc}^\infty (\mathcal {O} ,\mathbb {R}^{m-1})\) , we have that \(\phi \)  is a distributional solution of
\((D^{\phi }_2\phi ,\dots , D^{\phi }_m\phi ) =w\)  for a suitable \(w\in \mathcal {L}^\infty _{loc} (\mathcal {O} , \mathbb {R}^{m-1})\)  such that \(w(a)=(D^{\phi }_2\phi (a),\dots , D^{\phi }_m\phi (a) )\)  \(\mathcal {L}^{m+n-1} \) -a.e. in \(\mathcal {O}\) .

By Theorem \(\ref {cmpssteo17}\)  we know that there exist \((\phi _h)_{h\in \mathbb {N}} \subset \infty (\mathcal {O} ,\mathbb {R})\)  such that for all \(\mathcal {O}^{\prime } \Subset \mathcal {O}\)  we have

\( \phi _h\)  uniformly converges to \( \phi \)  in \(\mathcal {O}^{\prime } \)

\(|(D^{ \phi _h}_2 \phi _h(a), \dots , D^{ \phi _h}_m \phi _h(a)) |_{\mathbb {R}^{m-1}} \le \Vert  (D^{ \phi }_2 \phi (a), \dots , D^{ \phi }_m \phi (a)) \Vert _{\mathcal {L}^\infty (\mathcal {O}^{\prime },\mathbb {R}^{m-1})}\)  \(\forall a\in \mathcal {O} ^{\prime }\) , \(h\in \mathbb {N}\)

for each \(j=2,\dots , m\) , we have \(D^{ \phi _h}_j \phi _h (a)\rightarrow D^{ \phi }_j \phi (a)\)  a.e. \(a\in \mathcal {O}^{\prime }\)

Let \(w_h =(w_{2,h}, \dots , w_{m,h}):=(D_2^{\phi _h} \phi _h, \dots ,D_m^{\phi _h} \phi _h)\) . Observe that for each \(h\in \mathbb {N}\)  and \(\zeta \in \infty _c (\mathcal {O} ,\mathbb {R})\)
\(\int _\mathcal {O} \phi _h \left( \,X_j \zeta +\phi _h \sum _{s=1}^n b_ {j1 }^{(s)} Y_s \zeta \right) \, d\mathcal {L}^{m+n-1} = - \int _\mathcal {O} w_{j,h} \zeta \, d \mathcal {L}^{m+n-1}\) 
for all \(j=2,\dots , m\) . Getting to the limit for \(h\rightarrow \infty \)  we have
\(\int _\mathcal {O} \phi \left( \,X_j \zeta +\phi \sum _{s=1}^n b_ {j1 }^{(s)} Y_s \zeta \right) \, d\mathcal {L}^{m+n-1} = - \int _\mathcal {O} w_{j} \zeta \, d \mathcal {L}^{m+n-1}.\) 
Hence \(\phi \)  is a distributional solution of \((D^{\phi }_2\phi ,\dots , D^{\phi }_m\phi ) =w\)  in \(\mathcal {O}\) .
Theorem 7.2 
Let \(\mathbb {G}:= (\mathbb {R}^{m+n}, \cdot , \delta _\lambda )\)  be a Carnot group of step 2 and \(, \) W\( the complementary subgroups defined in (\ref {5.2.0}). Let \) :O  be a continuous map where \(\hat{\mathcal {O}} \)  is open in \(\mathbb {W}\)  and \(\phi :\mathcal {O} \rightarrow \mathbb {R}\)  is the map associated to \(\hat{\phi }\)  as in \((\ref {phipsi})\) . We also assume that

\(\phi \)  is a continuous distributional solution of \((D^{\phi }_2\phi ,\dots , D^{\phi }_m\phi ) =w\)  in \(\mathcal {O} \)  with \(w\in \mathcal {L}^\infty _{loc}(\mathcal {O} ,\mathbb {R}^{m-1})\)

\(\phi \)  is locally \(1/2 \) -Hölder continuous along the vertical components with Hölder's constant \(C_h>0\)

Then \(\hat{\phi }\)  is locally intrinsic Lipschitz function.

The proof of Theorem REF  relies on a preliminary result about the distributional solutions of the problem \((D^{\phi }_2\phi ,\dots , D^{\phi }_m\phi ) =w\)  in \(\mathcal {O} \) .
It is convenient to introduce the following notation: for \(j=2,\dots ,m \)  and
\(\hat{x}_j=(\hat{x}_2, \dots , \hat{x}_{j-1} ,\hat{x}_{j+1}, \dots , \hat{x}_{m}) \in \mathbb {R}^{m-2}\)  fixed, we denote by \((t,\hat{x}_j) :=(\hat{x}_2, \dots , \hat{x}_{j-1}, t ,\hat{x}_{j+1}, \dots , \hat{x}_{m} )\)
Lemma 7.3 
Under the same assumptions of Theorem \(\ref {lemma5.4bcsc}\) , for all \(j=2,\dots , m\)  there is \(C=C(\Vert  w_j\Vert _{\mathcal {L}^\infty (\mathcal {O}, \mathbb {R})}, C_h, b_ {j1 }^{(1)},\dots , b_ {j1 }^{(n)})>0\)  such that \(\phi \)  is a C-Lipschitz map along any characteristic line \(\gamma _{j}=(\gamma _{j1}, \dots , \gamma _{jn}):[-T, T ]\rightarrow \mathbb {R}^n\)  satisfying
\(\dot{\gamma }_{js} (t)= b_ {j1 }^{(s)} \phi (t,\hat{x}_j, \gamma _j(t))+ \frac{1}{2}\sum _{l=2}^{m}b^{(s)}_{jl}(\hat{x}_j )_l \quad \mbox{ for all } s=1,\dots , n\) 
with \(t\in [-T , T ]\)  and \(\hat{x}_j\in \mathbb {R}^{m-2}\)  fixed.

Fix \(j=2,\dots ,m\)  and \(\hat{x}_j \in \mathbb {R}^{m-2}\) .
Let \(t_1,t_2 \in (-T,T)\)  with \(t_1<t_2\)  and let \(\epsilon >0\) . In the similar way in [44]} and in [14]}, Lemma 5.1 we consider the functions \(\xi _1(x,y_1), \dots , \xi _n(x,y_n)\)  and \(h(x)\)  defined for small \(\delta >0\)  by
\(\xi _s (x,y_s)= \left\lbrace \begin{array}{l}\quad \quad 0\hspace{91.04872pt} -T<x<T, \, \, -r_s <y_s\le \gamma _s(x) -\epsilon -\delta \\\frac{1}{\delta }(y_s -\gamma _s(x)+\epsilon +\delta ) \hspace{18.49411pt} -T<x<T, \, \, \gamma _s (x) -\epsilon -\delta <y_s\le \gamma _s (x) -\epsilon \\\quad \quad 1 \hspace{91.04872pt} -T<x<T, \, \, \gamma _s(x) -\epsilon <y_s\le \gamma _s(x)\\\frac{1}{\delta }(-y_s +\gamma _s(x) +\delta ) \hspace{28.45274pt} -T<x<T, \, \, \gamma _s (x) <y_s\le \gamma _s(x) +\delta \\\quad \quad 0 \hspace{91.04872pt} -T<x<T, \, \, \gamma _s(x) +\delta <y_s< r_s\end{array}\right.\) 
for \(s=1,\dots ,n\)  and
\(h(x) = \left\lbrace \begin{array}{l}\quad \quad 0\hspace{71.13188pt} -T< x \le t_1 -\delta \\\frac{1}{\delta }(x-t_1 +\delta ) \hspace{36.98866pt} t_1 -\delta <x\le t_1\\\quad \quad 1 \hspace{72.55461pt} t_1 <x\le t_2\\\frac{1}{\delta }(-x+t_2 +\delta ) \hspace{29.87547pt} t_2<x \le t_2+\delta \\\quad \quad 0 \hspace{73.97733pt}t_2+\delta <x<T\end{array}\right.\) 
If \(\phi \)  is distributional solution of \((D_2^\phi \phi , \dots , D_m^\phi \phi )=w\)  in \(\mathcal {O} \) , for all test function \(\varphi \)  we have
\(\begin{aligned}\int _{-T}^T \int _{-r_1}^{r_1} \dots \int _{-r_n}^{r_n} (\phi (x,\hat{x}_j,y)\varphi _x(x,y) & +f_1(\phi (x,\hat{x}_j,y)) \varphi _{y_1}(x,y)+\dots +f_n(\phi (x,y)) \varphi _{y_n}(x,y)\\& +w_j(x,\hat{x}_j,y)\varphi (x,y))\, dx dy =0\end{aligned}\) 
where
\(\begin{aligned}f_s(\phi (x_j,\hat{x}_j,y))& = \frac{1}{2} \biggl ( b_{j1}^{(s)}\phi ^2(x_j,\hat{x}_j,y)+\phi (x_j,\hat{x}_j,y)\sum _{l=2}^{m}b_{jl}^{(s)}(\hat{x}_j)_l \biggl )\end{aligned}\) 
for all \((x_j,\hat{x}_j,y)\in \mathcal {O} \)  and \(s=1,\dots ,n\) .
We choose the test function \(\varphi (x,y_1,\dots ,y_n )= \xi _1 (x,y_1) \dots \xi _n (x,y_n) h(x)\)  and compute the limit \(\delta \rightarrow 0^+\) :
\(\begin{aligned}\lim _{\delta \rightarrow 0^+} \int _{-T}^T \int _{-r_1}^{r_1} \dots \int _{-r_n}^{r_n} (\phi \varphi _x+f_1(\phi ) ( \xi _1)_{y_1} \xi _2 \dots \xi _n h+\dots +f_n(\phi ) \xi _1 \dots (\xi _n)_{y_n}h+w_j\xi _1 \dots \xi _nh)\, dx dy\end{aligned}\) 
For simplicity we consider the case \(n=2\) . We compute the following limit:
\(\begin{aligned}\lim _{\delta \rightarrow 0^+} & \int _{-T}^T \int _{-r_1}^{r_1} \int _{-r_2}^{r_2} \biggl (\phi \biggl ((\xi _1)_x\xi _2 h+\xi _1(\xi _2)_x h+\xi _1\xi _2 h_x\biggl )+f_1(\phi ) ( \xi _1)_{y_1} \xi _2 h \\& \quad +f_2(\phi ) \xi _1(\xi _2)_{y_2}h + w_j\xi _1 \xi _2h\biggl )\, dx dy_1dy_2=: \lim _{\delta \rightarrow 0^+} I_1+I_2+I_3+I_4+I_5+I_6.\end{aligned}\) 
Then
\(\begin{aligned}\lim _{\delta \rightarrow 0^+} I_1& = \lim _{\delta \rightarrow 0^+} \int _{-T}^T \int _{-r_1}^{r_1} \int _{-r_2}^{r_2} \phi (\xi _1)_x\xi _2 h\, dx dy_1dy_2\\& =\int _a^b \int _{\gamma _2(x)-\epsilon }^{\gamma _2(x)} \dot{\gamma }_1(x) \Bigl (\phi (x,\hat{x}_j,\gamma _1(x), y_2) - \phi (x,\hat{x}_j,\gamma _1 (x)-\epsilon , y_2)\Bigl )\, dxdy_2\\\lim _{\delta \rightarrow 0^+} I_2& = \lim _{\delta \rightarrow 0^+} \int _{-T}^T \int _{-r_1}^{r_1} \int _{-r_2}^{r_2} \phi \xi _1(\xi _2)_x h\, dx dy_1dy_2\\& =\int _{t_1}^{t_2} \int _{\gamma _1(x)-\epsilon }^{\gamma _1(x)} \dot{\gamma }_2(x) \Bigl (\phi (x,\hat{x}_j,y_1, \gamma _2(x)) - \phi (x,\hat{x}_j,y_1, \gamma _2 (x)-\epsilon )\Bigl )\, dxdy_1\\\end{aligned}\) 
\(\begin{aligned}\lim _{\delta \rightarrow 0^+} I_3& = \lim _{\delta \rightarrow 0^+} \int _{-T}^T \int _{-r_1}^{r_1} \int _{-r_2}^{r_2} \phi \xi _1\xi _2 h_x\, dx dy_1dy_2\\& = \int _{\gamma _1(t_1)-\epsilon }^{\gamma _1(t_1)} \int _{\gamma _2(t_1)-\epsilon }^{\gamma _2(t_1)} \phi (t_1,\hat{x}_j,y_1, y_2) \, dy_1dy_2 -\int _{\gamma _1(t_2)-\epsilon }^{\gamma _1(t_2)} \int _{\gamma _2(t_2)-\epsilon }^{\gamma _2(t_2)} \phi (t_2,\hat{x}_j,y_1, y_2) \, dy_1dy_2\\\lim _{\delta \rightarrow 0^+} I_4& = \lim _{\delta \rightarrow 0^+} \int _{-T}^T \int _{-r_1}^{r_1} \int _{-r_2}^{r_2} f_1(\phi ) ( \xi _1)_{y_1} \xi _2 h\, dx dy_1dy_2\\& = \int _{t_1}^{t_2} \int _{\gamma _2(x)-\epsilon }^{\gamma _2(x)} \Bigl ( f_1(\phi (x,\hat{x}_j,\gamma _1 (x)- \epsilon ,y_2))-f_1(\phi (x,\hat{x}_j,\gamma _1 (x) ,y_2)) \Bigl )\, dxdy_2\\\lim _{\delta \rightarrow 0^+} I_5& = \lim _{\delta \rightarrow 0^+} \int _{-T}^T \int _{-r_1}^{r_1} \int _{-r_2}^{r_2} f_2(\phi )\xi _1 ( \xi _2)_{y_2} h\, dx dy_1dy_2\\& = \int _{t_1}^{t_2} \int _{\gamma _1(x)-\epsilon }^{\gamma _1(x)} \Bigl ( f_2(\phi (x,y_1, \gamma _2 (x)- \epsilon ))-f_2(\phi (x,y_1, \gamma _2 (x) )) \Bigl )\, dxdy_1\\\lim _{\delta \rightarrow 0^+} I_6 &= \lim _{\delta \rightarrow 0^+} \int _{-T}^T \int _{-r_1}^{r_1} \int _{-r_2}^{r_2}w_j\xi _1 \xi _2h\, dx dy_1dy_2 = \int _{t_1}^{t_2} \int _{\gamma _1(x)-\epsilon }^{\gamma _1(x)}\int _{\gamma _2(x)-\epsilon }^{\gamma _2(x)} w_j\, dxdy_1dy_2\end{aligned}\) 
Hence we obtain
\(\begin{aligned}& \int _{\gamma _1(t_2)-\epsilon }^{\gamma _1(t_2)} \int _{\gamma _2(t_2)-\epsilon }^{\gamma _2(t_2)} \phi (t_2,\hat{x}_j,y_1, y_2) \, dy_1dy_2 - \int _{\gamma _1(t_1)-\epsilon }^{\gamma _1(t_1)} \int _{\gamma _2(t_1)-\epsilon }^{\gamma _2(t_1)} \phi (t_1,\hat{x}_j,y_1, y_2) \, dy_1dy_2\\& \quad - \int _{t_1}^{t_2} \int _{\gamma _1(x)-\epsilon }^{\gamma _1(x)}\int _{\gamma _2(x)-\epsilon }^{\gamma _2(x)} w_j (x,\hat{x}_j,y_1, y_2) \, dxdy_1dy_2\\& = \int _{t_1}^{t_2} \int _{\gamma _2(x)-\epsilon }^{\gamma _2(x)} f_1(\phi (x,\hat{x}_j,\gamma _1 (x)- \epsilon ,y_2))-f_1(\phi (x,\hat{x}_j,\gamma _1 (x) ,y_2))\\& \quad + \dot{\gamma }_1(x) \Bigl (\phi (x,\hat{x}_j,\gamma _1(x), y_2) - \phi (x,\hat{x}_j,\gamma _1 (x)-\epsilon , y_2)\Bigl )\, dx dy_2 \\& \quad + \int _{t_1}^{t_2} \int _{\gamma _1(x)-\epsilon }^{\gamma _1(x)} f_2(\phi (x,\hat{x}_j,y_1, \gamma _2 (x)- \epsilon ))-f_2(\phi (x,\hat{x}_j,y_1, \gamma _2 (x) )) \\&\quad + \dot{\gamma }_2(x) \Bigl (\phi (x,\hat{x}_j,y_1, \gamma _2(x)) - \phi (x,\hat{x}_j,y_1, \gamma _2 (x)-\epsilon )\Bigl )\, dxdy_1\end{aligned}\) 
From (REF ) with \(n=2\)  we deduce that
\(\begin{aligned}& \int _{\gamma _1(t_2)-\epsilon }^{\gamma _1(t_2)} \int _{\gamma _2(t_2)-\epsilon }^{\gamma _2(t_2)} \phi (t_2,\hat{x}_j,y_1, y_2) \, dy_1dy_2 - \int _{\gamma _1(t_1)-\epsilon }^{\gamma _1(t_1)} \int _{\gamma _2(t_1)-\epsilon }^{\gamma _2(t_1)} \phi (t_1,\hat{x}_j,y_1, y_2) \, dy_1dy_2\\& \quad - \int _{t_1}^{t_2} \int _{\gamma _1(x)-\epsilon }^{\gamma _1(x)}\int _{\gamma _2(x)-\epsilon }^{\gamma _2(x)} w_j (x,\hat{x}_j,y_1, y_2) \, dxdy_1dy_2\\& = \int _{t_1}^{t_2} \int _{\gamma _2(x)-\epsilon }^{\gamma _2(x)} \frac{1}{2}b_{j1}^{(1)}\biggl ( \phi (x,\hat{x}_j,\gamma _1(x)-\epsilon ,y_2)- \phi (x,\hat{x}_j,\gamma _1(x) ,y_2) \biggl ) \biggl ( \phi (x,\hat{x}_j,\hat{x}_j,\gamma _1(x),y_2)+\\& + \phi (x,\hat{x}_j,\hat{x}_j,\gamma _1(x)-\epsilon ,y_2) -2\phi (x,\hat{x}_j,\hat{x}_j,\gamma _1(x),\gamma _2(x)) \biggl ) \, dxdy_2 + \int _{t_1}^{t_2} \int _{\gamma _1(x)-\epsilon }^{\gamma _1(x)} \frac{1}{2}b_{j1}^{(2)}\biggl ( \phi (x,\hat{x}_j,y_1,\gamma _2(x)-\epsilon )\\ & - \phi (x,y_1,\gamma _2(x) ) \biggl ) \biggl ( \phi (x,y_1,\gamma _2(x)) + \phi (x,y_1, \gamma _2(x)-\epsilon ) -2\phi (x,\hat{x}_j,\gamma _1(x),\gamma _2(x)) \biggl ) \, dxdy_1\\\end{aligned}\) 
If \(b_{j1}^{(1)}, b_{j1}^{(2)} = 0\)  the integral (REF ) is zero. On the other hand if \(b_{j1}^{(1)} , b_{j1}^{(2)}\ne 0\) , because \(\phi \)  is locally \(1/2 \) -Hölder continuous along the vertical components with Hölder's constant \(C_h>0\)
\(\begin{aligned}&\frac{|\phi (x,\hat{x}_j,\gamma _1(x),y_2)- \phi (x,\hat{x}_j,\gamma _1(x)-\epsilon ,y_2)| }{\sqrt{\epsilon }} \le C_h\\& \frac{|\phi (x,\hat{x}_j,\gamma _1(x),y_2) -\phi (x,\hat{x}_j,\gamma _1(x),\gamma _2(x)) |}{\sqrt{\epsilon }} \le C_h\\& \frac{ |\phi (x,\hat{x}_j,\gamma _1(x)-\epsilon ,y_2) -\phi (x,\hat{x}_j,\gamma _1(x),\gamma _2(x)) |}{\sqrt{2\epsilon }} \le C_h\end{aligned}\) 
for \(x\in [t_1,t_2]\)  and \(y_2\in (\gamma _2(x)-\epsilon , \gamma _2(x))\) . Hence dividing by \(\epsilon ^2\)  in \((\ref {lipottobre})\)  and getting to the limit to \(\epsilon \rightarrow 0\) , we obtain
\(\begin{aligned}&\lim _{\epsilon \rightarrow 0 } \frac{1}{\epsilon ^2} \int _{t_1}^{t_2} \int _{\gamma _2(x)-\epsilon }^{\gamma _2(x)} \frac{1}{2}b_{j1}^{(1)}\biggl ( \phi (x,\hat{x}_j,\gamma _1(x)-\epsilon ,y_2)- \phi (x,\hat{x}_j,\gamma _1(x) ,y_2) \biggl ) \biggl ( \phi (x,\hat{x}_j,\gamma _1(x),y_2)+\\& \quad + \phi (x,\hat{x}_j,\gamma _1(x)-\epsilon ,y_2) -2\phi (x,\hat{x}_j,\gamma _1(x),\gamma _2(x)) \biggl ) \, dxdy_2 \\& \le |b_{j1}^{(1)}| \frac{(1+\sqrt{2})C_h^2}{2} (t_2-t_1).\end{aligned}\) 
Moreover
\(\begin{aligned}&\lim _{\epsilon \rightarrow 0 } \frac{1}{\epsilon ^2} \int _{t_1}^{t_2} \int _{\gamma _1(x)-\epsilon }^{\gamma _1(x)} \frac{1}{2}b_{j1}^{(2)}\biggl ( \phi (x,\hat{x}_j,y_1,\gamma _2(x)-\epsilon )- \phi (x,\hat{x}_j,y_1,\gamma _2(x) ) \biggl ) \biggl ( \phi (x,\hat{x}_j,y_1,\gamma _2(x))+\\& \quad + \phi (x,\hat{x}_j,y_1, \gamma _2(x)-\epsilon ) -2\phi (x,\hat{x}_j,\gamma _1(x),\gamma _2(x)) \biggl ) \, dxdy_1\\& \le |b_{j1}^{(2)} | \frac{(1+\sqrt{2})C_h^2}{2} (t_2-t_1),\end{aligned}\) 
where in the last inequality we have used the fact
\(\begin{aligned}&\frac{|\phi (x,\hat{x}_j,y_1,\gamma _2(x)-\epsilon )- \phi (x,\hat{x}_j,y_1,\gamma _2(x)) |}{\sqrt{\epsilon }} \le C_h\\& \frac{|\phi (x,\hat{x}_j,y_1,\gamma _2(x)) -\phi (x,\hat{x}_j,\gamma _1(x),\gamma _2(x)) |}{\sqrt{\epsilon }} \le C_h\\& \frac{ |\phi (x,\hat{x}_j,y_1,\gamma _2(x)-\epsilon ) -\phi (x,\hat{x}_j,\gamma _1(x),\gamma _2(x)) |}{\sqrt{2\epsilon }} \le C_h\end{aligned}\) 
for \(x\in [t_1,t_2]\)  and \(y_1\in (\gamma _1(x)-\epsilon , \gamma _1(x))\) .
Now putting together \((\ref {dise01})\) , \((\ref {dise02})\)  and \((\ref {dise03})\) , dividing by \(\epsilon ^2\)  and getting to the limit to \(\epsilon \rightarrow 0\) , we obtain
\(\phi (t_2,\hat{x}_j , \gamma _j(t_2)) -\phi (t_1,\hat{x}_j , \gamma _j(t_1 )) \le \biggl (\Vert w_j\Vert _{\mathcal {L}^\infty (\mathcal {O} ,\mathbb {R})} + \frac{(1+\sqrt{2})C_h^2}{2} \Big (|b_{j1}^{(1)} |+ |b_{j1}^{(2)}|\Big ) \biggl ) (t_2-t_1)\) 
for \(t_1,t_2 \in (-T,T)\)  with \(t_1<t_2\) .
In the similar way, using again (REF ) and (REF ) we have
\(\phi (t_2,\hat{x}_j , \gamma _j(t_2)) - \phi (t_1,\hat{x}_j , \gamma _j(t_1)) \ge - \biggl (\Vert w_j\Vert _{\mathcal {L}^\infty (\mathcal {O},\mathbb {R})} + \frac{(1+\sqrt{2})C_h^2}{2} \Big (|b_{j1}^{(1)} |+ |b_{j1}^{(2)}|\Big ) \biggl ) (t_2-t_1)\) 
for \(t_1,t_2 \in (-T,T)\)  with \(t_1<t_2\) .
Hence combining (REF ) and (REF ) we get
\(|\phi (t_2,\hat{x}_j , \gamma _j(t_2 )) - \phi (t_1,\hat{x}_j , \gamma _j(t_1)) | \le \biggl (\Vert w_j\Vert _{\mathcal {L}^\infty (\mathcal {O},\mathbb {R})} + \frac{(1+\sqrt{2})C_h^2}{2} \Big (|b_{j1}^{(1)} |+ |b_{j1}^{(2)}|\Big ) \biggl ) (t_2-t_1),\) 
for \(t_1,t_2 \in (-T , T)\)  with \(t_1<t_2\) , i.e. \(\phi \)  is a Lipschitz map along any characteristic line \(\gamma _{j}\)  as desired.
Example 7.1 We consider free step 2 group \(\mathbb {F}_{m,2}\) . As we said in Remark REF , the composition law \((\ref {1})\)  is given by \(\mathcal {B}^{(s)}\equiv \mathcal {B}^{(l,h)}\)  where \(1\le h< l\le m\)  and \(\mathcal {B}^{(l,h)}\)  has entries \(-1\)  in position \((l,h)\) , 1 in position \((h,l)\)  and 0 everywhere else.
For each \(j=2,\dots ,m\)  and \(\hat{x}_j\in \mathbb {R}^{m-2}\)  fixed the characteristic line \(\gamma _{j}=(\gamma _{j1}, \dots , \gamma _{jn}):[-\delta , \delta ]\rightarrow \mathbb {R}^n\)  has the following form:
\(\begin{aligned}\dot{\gamma }_{js_1} (t)&= b_ {j1 }^{(s_1)} \phi (t,\hat{x}_j, \gamma _j(t))= - \phi (t,\hat{x}_j, \gamma _j(t)) \\\dot{\gamma }_{js_2} (t)&= \frac{1}{2}b^{(s_2)}_{jh_2}(\hat{x}_j )_{h_2}= - \frac{1}{2}(\hat{x}_j )_{h_2}\\\dot{\gamma }_{js_3} (t)&= 0\end{aligned}\) 
where \(\mathcal {B}^{(s_i)}\equiv \mathcal {B}^{(l_i,h_i)}\)  is such that

\(s_1\)  is the unique index \((l_1,h_1)=(j,1)\) .

\(s_2\)  is such that \(h_2<l_2=j\)  and \(h_2\ne 1\) .

\(s_3\)  represents the other cases.


Now we are able to show the proof of Theorem REF .
Fix \(b\in \mathcal {O}\) . According to Remark REF , we would like to prove that there exists \(\hat{C}>0\)  such that
\(|\phi (a^{\prime })-\phi (a)| \le \hat{C} \Vert \hat{\phi }(i(a))^{-1}i(a)^{-1}i(a^{\prime })\hat{\phi }(i(a)) \Vert  \qquad \mbox{for all } \, a, a^{\prime } \in \mathcal {U}(b,\delta )\) 
with \(\delta >0\) . Let \(a=( x,y),a^{\prime }= ( x^{\prime },y^{\prime })\)  points of \(\mathcal {O}\)  be sufficiently close to \(b\) , and let \(\bar{D}_j\)  be the vector fields given by \(\bar{D}_j=(x^{\prime }_j-x_j)D^\phi _j\)  for \(j\in \lbrace 2,\dots ,m\rbrace \) . We define
\(\begin{aligned}a_1 & :=a\\a_2 & :=\exp (\bar{D} _2)(a_1)\\a_3 & :=\exp (\bar{D} _3)(a_2)\\\vdots & \\a_m & :=\exp (\bar{D}_m)(a_{m-1}).\end{aligned}\) 
More precisely for \(j\in \lbrace 2,\dots ,m\rbrace \)
\(a_j=(x^{\prime }_2,\dots ,x^{\prime }_j,x_{j+1},\dots ,x_m,y^{a_j})\) 
with
\(\begin{aligned}y^{a_j}_s & = y_s+\sum _{l=2}^{j} \left( b_{l1}^{(s)} \int _0^{x^{\prime }_l-x_l} \phi \Big (\exp (rD^\phi _l (a_{l-1}))\Big )\, dr +\frac{1}{2}(x^{\prime }_l-x_l)\Big (\sum _{ h=2 }^{l} x^{\prime }_h b_{lh}^{(s)}+\sum _{ h=l+1}^{m} x_h b_{lh}^{(s)}\Big ) \right)\\& = y^{a_{j-1}}_s +\left( b_{j1}^{(s)} \int _0^{x^{\prime }_j-x_j} \phi \Big (\exp (rD^\phi _j (a_{j-1}))\Big )\, dr +\frac{1}{2}(x^{\prime }_j-x_j)\Big (\sum _{ h=2 }^{j} x^{\prime }_h b_{jh}^{(s)}+\sum _{ h=j+1}^{m} x_hb_{jh}^{(s)}\Big ) \right)\end{aligned}\) 
for \(s=1,\dots ,n\) . By Lemma REF  \((-\delta ,\delta )\ni r\mapsto \phi \Big (\exp (rD^\phi _j (a_{j-1}))\Big )\)  is Lipschitz for all \(j=2,\dots , m\)  and so \(a_2,\dots ,a_m \)  are well defined if \(a,a^{\prime }\in \mathcal {U}(b,\delta )\)  for a sufficiently small \(\delta \) .
We observe that
\(|\phi (a^{\prime })-\phi (a)| \le |\phi (a^{\prime })-\phi (a_m)| + \sum _{l=2}^{m} |\phi (a_l)-\phi (a_{l-1} )|\) 
and by Lemma \(\ref {lemma5.1bcsc} \)  there is \(C>0\)  such that
\(\sum _{l=2}^{m} |\phi (a_l)-\phi (a_{l-1} )| \le \sum _{l=2}^{m} C | x^{\prime }_l - x_l |_{\mathbb {R}^{m-1}} \le C \Vert \hat{\phi }(i(a))^{-1}i(a)^{-1}i(a^{\prime })\hat{\phi }(i(a)) \Vert .\) 
Hence in order to establish \((\ref {lipsch})\)  we show that there exists \(C_1>0\)  such that
\(|\phi (a^{\prime })-\phi (a_m)|\le C_1 \Vert \hat{\phi }(i(a))^{-1}i(a)^{-1}i(a^{\prime })\hat{\phi }(i(a)) \Vert .\) 
Recalling that \(a_m=(x^{\prime },y^{a_m})\)  and using \(\phi \)  is locally \(1/2 \) -Hölder continuous along the vertical components with Hölder's constant \(C_h>0\) , it follows
\(|\phi (a^{\prime })-\phi (a_m)|\le C_h |y^{\prime }-y^{a_m}|^{1/2}_{\mathbb {R}^{n}}\) 
Moreover arguing as in the proof of the implication \(4. \Rightarrow 2.\)  in Theorem 5.7 in [10]}, we have
\(\begin{aligned}|y^{\prime }-y^{a_m}|_{\mathbb {R}^{n}} & \le \sum _{ s=1}^{n} \biggl |y^{\prime }_s-y_s+\sum _{l=2}^{m} \biggl ( b_{1l}^{(s)} \int _0^{x^{\prime }_l-x_l} \phi \Big (\exp (rD^\phi _l (a_{l-1}))\Big )\, dr +\\&\quad -\frac{1}{2}(x^{\prime }_l-x_l)\Big (\sum _{ i=2 }^{l} x^{\prime }_i b_{li}^{(s)}+\sum _{ i=l+1}^{m} x_i b_{li}^{(s)}\Big ) \biggl ) \biggl | \\& \le \sum _{ s=1}^{n} \biggl |y^{\prime }_s- y_s +\phi (a)\sum _{ l=2 }^{m} (x^{\prime }_l-x_l) b_{1l}^{(s)} -\frac{1}{2}\langle \mathcal {B}^{(s)} x, x^{\prime }- x \rangle \biggl |\\& \quad + \sum _{ s=1}^{n} \biggl | -\frac{1}{2} \sum _{l=2}^{m}(x^{\prime }_l-x_l)\Big (\sum _{ i=2 }^{l} x^{\prime }_i b_{li}^{(s)}+\sum _{ i=l+1}^{m} x_i b_{li}^{(s)}\Big ) + \frac{1}{2} \langle \mathcal {B}^{(s)} x,x^{\prime }- x \rangle \biggl | \\& \quad +\sum _{ s=1}^{n} \biggl | -\phi (a)\sum _{ l=2 }^{m} (x^{\prime }_l-x_l) b_{1l}^{(s)} +\sum _{l=2}^{m} b_{1l}^{(s)} \int _0^{x^{\prime }_l-x_l} \phi \Big (\exp (r D^\phi _l (a_{l-1}))\Big )\, dr \biggl | \\& \le c_1 \Vert \hat{\phi }(i(a))^{-1}i(a)^{-1}i(a^{\prime })\hat{\phi }(i(a)) \Vert ^2 + \frac{1}{2}n\mathcal {B}_M|x^{\prime }-x|^2_{\mathbb {R}^{m-1}} +\\& \quad +\sum _{ s=1}^{n} \biggl | -\phi (a)\sum _{ l=2 }^{m} (x^{\prime }_l-x_l) b_{1l}^{(s)} +\sum _{l=2}^{m} b_{1l}^{(s)} \int _0^{x^{\prime }_l-x_l} \phi \Big (\exp (r D^\phi _l (a_{l-1}))\Big )\, dr \biggl | \\\end{aligned}\) 
where \(c_1 \)  is given by (REF ) and \(\mathcal {B}_{M} = \max \lbrace  b_{ij}^{(s)} \, | \, i,j=1,\dots , m \, , s=1,\dots , n \rbrace \) .
Note that we have used
\(\begin{aligned}\frac{1}{2} \langle \mathcal {B}^{(s)} x, x^{\prime } & - x\rangle -\frac{1}{2} \sum _{l=2}^{m} (x^{\prime }_l -x_l)\Big (\sum _{ i=2 }^{l} x^{\prime }_i b_{li}^{(s)} +\sum _{ i=l+1}^{m} x_i b_{li}^{(s)}\Big ) \\&= -\frac{1}{2} \sum _{l=2}^{m} (x^{\prime }_l-x_l)\Big (\sum _{ i=2 }^{l} x^{\prime }_i b_{li}^{(s)}+\sum _{ i=l+1}^{m} x_i b_{li}^{(s)} -\sum _{ i=2}^{m} x_i b_{li}^{(s)} \Big ) \\& \le \frac{1}{2}n\mathcal {B}_M|x^{\prime }-x|^2_{\mathbb {R}^{m-1}} .\end{aligned}\) 
Finally, the last term
\(\begin{aligned}& \sum _{ s=1}^{n} \biggl | -\phi (a)\sum _{ l=2 }^{m} (x^{\prime }_l-x_l) b_{1l}^{(s)}+\sum _{l=2}^{m} b_{1l}^{(s)} \int _0^{x^{\prime }_l-x_l} \phi \Big (\exp (r D^\phi _l (B_{l-1}))\Big )\, dr \biggl | \\&\qquad \le R_1(a,a^{\prime })+R_2(a,a^{\prime })\end{aligned}\) 
where
\(\begin{aligned}R_1(a,a^{\prime })& := \sum _{ s=1}^{n} \sum _{l=2}^{m}\biggl | b_{1l}^{(s)} \int _0^{x^{\prime }_l-x_l} \phi \Big (\exp (rD^\phi _l (a_{l-1}))\Big )\, dr - b_{1l}^{(s)}\phi (a_{l-1})(x^{\prime }_l-x_l) \biggl |\\R_2(a,a^{\prime }) &:= \sum _{ s=1}^{n} \biggl | \sum _{l=2}^{m} b_{1l}^{(s)}(x^{\prime }_l-x_l) \Big ( \phi (a_{l-1})-\phi (a)\Big ) \biggl | \\\end{aligned}\) 
We would like to show that there exist \(C_1, C_2 >0\)  such that
\(R_1(a,a^{\prime }) \le C_1 |x^{\prime }-x|^2_{\mathbb {R}^{m-1}}\) 
\(R_2(a,a^{\prime }) \le C_2 |x^{\prime }-x|^2_{\mathbb {R}^{m-1}}\) 
for all \(a,a^{\prime }\in \mathcal {U}(b,\delta )\)  and consequently
\(|y^{\prime }-y^{a_m}|_{\mathbb {R}^{n}} \le c_1 \Vert \hat{\phi }(i(a))^{-1}i(a)^{-1}i(a^{\prime })\hat{\phi }(i(a)) \Vert ^2 + \left( \frac{1}{2}n\mathcal {B}_M +C_1+C_2 \right) |x^{\prime }-x|^2_{\mathbb {R}^{m-1}}\) 
Hence there is \( C_3>0\)  such that
\(|y^{\prime }-y^{a_m}|^{1/2} _{\mathbb {R}^{n}} \le C_3 \Vert \hat{\phi }(i(a))^{-1}i(a)^{-1}i(a^{\prime })\hat{\phi }(i(a)) \Vert ,\) 
i.e. (REF ) is true.
We start to consider \(R_1(a,a^{\prime })\) . Fix \(l=2,\dots , m\) . For \(t\in [-\delta , \delta ]\)  we define
\(g_l (t):=\sum _{s=1}^{n} b_{1l}^{(s)} \biggl (\int _0^t \phi (\exp (rD^\phi _l)(a_{l-1}))\, dr - t \phi (a_{l-1}) \biggl )\) 
Observe that
\(\begin{aligned}b_{1l}^{(s)}\int _0^t \left( \phi (\exp (rD^\phi _l)(a_{l-1})) - \phi (a_{l-1}) \right) \, dr = O(t^2)\end{aligned}\) 
and so there is \(C_l>0\)  such that
\(|g_l (t)| \le C_l t^2, \hspace{14.22636pt} \forall t\in [-\delta , \delta ]\) 
Hence set \(t=x^{\prime }_l-x_l\)  we get
\(|g_l (x^{\prime }_l-x_l)| \le C_l (x^{\prime }_l-x_l)^2\) 
and consequently \((\ref {r1})\)  follows from
\(\sum _{l=2}^{m} |g_l (x^{\prime }_l-x_l )| \le \sum _{l=2}^{m} C_l (x^{\prime }_l-x_l)^2 \le C_1|x^{\prime }-x|_{\mathbb {R}^{m-1}} ^2.\) 
Now we consider \(R_2(a,a^{\prime })\) . Observe that
\(\begin{aligned}\sum _{ s=1}^{n} \biggl | \sum _{l=2}^{m} b_{1l}^{(s)}(x^{\prime }_l& -x_l) \Big ( \phi (a_{l-1})-\phi (a)\Big ) \biggl | \le n \mathcal {B}_M \sum _{l=2}^{m} |x^{\prime }_l-x_l| \left| \phi (a_{l-1})-\phi (a) \right| \\& \le n \mathcal {B}_M \sum _{l=2}^{m} |x^{\prime }_l-x_l| \Big (\sum _{h=2}^{l-1}|\phi (a_h)-\phi (a_{h-1})| \Big )\\& \le n \mathcal {B}_M \sum _{l=2}^{m} |x^{\prime }_l-x_l| \Big ( \sum _{h=2}^{l-1}\Big | \int _0^1 (\bar{D}_h \phi )(\exp (r \bar{D}_h (a_{h-1})))\, dr \Big |\Big ) \\& \le n \mathcal {B}_M \sum _{l=2}^{m} |x^{\prime }_l-x_l| \Big (\sum _{h=2}^{l-1} \Big |(x^{\prime }_h-x_h) \left(D^\phi _h \phi ( a_{h-1} )+o(1) \right) \Big | \Big )\\& \le n \mathcal {B}_M C |x^{\prime }-x|^2_{\mathbb {R}^{m-1}}\end{aligned}\) 
Then \((\ref {r2})\)  follows with \( C_2:=n \mathcal {B}_M C \)  and (REF ) is true.
Finally putting together \((\ref {rimanente5})\) , \((\ref {rimanente4})\)  and \((\ref {rimanenteok1})\) , \((\ref {lipsch})\)  holds and the proof is complete.

A characterization of \(\mathbb {G}\) -regular hypersurfaces
In this section we will prove a new characterization of \(\mathbb {G}\) -regular hypersurfaces (see Definition REF ). Here we consider the non linear first order system
\(\left(D^{\phi }_2\phi ,\dots , D^{\phi }_m\phi \right)= w\) 
where \(w:\mathcal {O}\rightarrow \mathbb {R}^{m-1}\)  is a given continuous function and not measurable as opposed to the previous sections.
In [10]}, we give some equivalent conditions about \(\mathbb {G}\) -regular hypersurface inside the Carnot groups of step 2 (see Theorem 5.7 in [10]}). In this section we give another one. More precisely, we show that if a continuous map \(\phi :\mathcal {O}\subset \mathbb {R}^{m+n-1}\rightarrow \mathbb {R}\)  is locally little \(1/2 \) -Hölder continuous along the vertical components, then \(\phi \)  is a distributional solution in an open set \(\mathcal {O}\)  of the non linear first order system \((\ref {sistema})\)  if and only if \(\hat{\phi }: \hat{\mathcal {O}}\rightarrow  is uniformly intrinsic differentiable in \) O\( and consequently its graph is a \) G\(-regular hypersurface. The main equivalence result is contained in Theorem \ref {CharacterizationRegularSurfaces}.\) In order to state the equivalence result we have to be precise about the meaning of being a solution of (REF ). To this aim we recall a notion of generalized solutions of systems of this kind. These generalized solutions, denoted
broad* solutions were introduced and studied for the system (REF ) inside Heisenberg groups in [21]}, [15]}. For a more complete bibliography we refer to the bibliography in [21]}. Our strategy will be to prove that each continuous distributional solution of the system (REF ) is a broad* solution and vice versa (see Proposition REF ).
Definition 8.1 
Let \(\mathcal {O} \subset \mathbb {R}^{m+n-1}\)  be open and \(w:=\left(w_2,\dots ,w_m\right):\mathcal {O} \rightarrow \mathbb {R}^{m-1}\)  a continuous function.
With the notations of Definition REF  we say that \(\phi \in \mathcal {O},\mathbb {R})\)  is a broad* solution in \(\mathcal {O}\)  of the system
\(\left(D^{\phi }_2\phi ,\dots , D^{\phi }_m\phi \right)= w \) 
if for every \(a_0\in \mathcal {O}\)  there are \(0< \delta _2 < \delta _1\)  and \(m-1\)  maps \(\exp _{a_0}(\cdot D^\phi _j)(\cdot )\)
\(\begin{split}\exp _{a_0}(\cdot D^\phi _j)(\cdot ):[-\delta _2,\delta _2]&\times \mathcal {I}(a_0,\delta _2)\rightarrow \mathcal {I}(a_0, \delta _1)\\(t,&a) \mapsto \exp _{a_0}(tD^\phi _j)(a )\end{split}\) 
for \(j=2,\dots , m\) , where \( \mathcal {I}(a_0, \delta ):= \mathcal {U}(a_0,\delta )\cap \mathbb {W}\)  and \(\mathcal {I}(a_0, \delta _1)\subset \mathcal {O}\) . Moreover these maps, called exponential maps of the vector fields \(D^\phi _2,\dots ,D^\phi _m\) , are required to have the following properties
\(t\mapsto \gamma ^j_a (t):=\exp _{a_0} (tD^\phi _j)(B) \in 1 ([-\delta _2,\delta _2], \mathbb {R}^{m+n-1})\) 
for all \(a\in \mathcal {I}(a_0,\delta _2)\)  and
\(\left\lbrace \begin{array}{l}\dot{\gamma }^j_a =D^\phi _j \circ \gamma ^j_a\\\gamma ^j_a (0)=a\end{array}\right.\) 
\(\phi (\gamma ^j_a (t))-\phi (\gamma ^j_a (0))=\int _0^t w_j (\gamma ^j_a (r))\, dr\) 
once more
for all \(a\in \mathcal {I}(a_0,\delta _2)\) .

Proposition 8.1 
Let \(\mathbb {G}:= (\mathbb {R}^{m+n}, \cdot , \delta _\lambda )\)  be a Carnot group of step 2 and \(, \) W\( the complementary subgroups defined in (\ref {5.2.0}). Let \) : O  be a continuous map, where \(\hat{ \mathcal {O}}\)  is an open subset of \(\mathbb {W}\)  and \(\phi :\mathcal {O} \rightarrow \mathbb {R}\)  is the map associated to \(\hat{\phi }\)  as in \((\ref {phipsi})\) . Then the following conditions are equivalent:

\(\phi \)  is locally little \(1/2 \) -Hölder continuous along the vertical components, i.e. that is
\(\phi \in \mathcal {O},\mathbb {R})\)  and for all \(\mathcal {O}^{\prime } \Subset \mathcal {O}\)  and \((x,y), (x,y^{\prime })\in \mathcal {O}^{\prime }\)
\(\lim _{r\rightarrow 0^+} \sup _{0<|y^{\prime }-y|_{\mathbb {R}^{n}} <r} \, \frac{|\phi (x,y^{\prime })-\phi (x,y)|}{|y^{\prime }-y |_{\mathbb {R}^{n}}^{1/2}} =0.\) 
and there exists \(w\in \mathcal {O} , \mathbb {R}^{m-1})\)  such that \(\phi \)  is a broad* solution of
\(\left(D^{\phi }_2\phi ,\dots , D^{\phi }_m\phi \right)= w, \quad \mbox{ in } \mathcal {O}\) 

\(\phi \)  is locally little \(1/2 \) -Hölder continuous along the vertical components and there exists \(w\in \mathcal {O} , \mathbb {R}^{m-1})\)  such that \(\phi \)  is distributional solution of
\(\left(D^{\phi }_2\phi ,\dots , D^{\phi }_m\phi \right)= w, \quad \mbox{ in } \mathcal {O}\) 


\(\mathbf {(1) \Rightarrow (2)}\) .
By Theorem \(5.7\)  in [10]}, we know that there is a family of functions \(\phi _\epsilon \in \mathbb {C}^1(\mathcal {O}, \mathbb {R})\)  such that for all \(\mathcal {O}^{\prime }\Subset \mathcal {O}\) ,
\(\phi _\epsilon \rightarrow \phi \quad and \quad D_j^{{\phi _\epsilon }} {\phi _\epsilon } \rightarrow D_j^{{\phi }} {\phi }\) 
for \(j=2,\dots , m\)  uniformly on \(\mathcal {O} ^{\prime }\)  as \(\epsilon \rightarrow 0^+\) . Then for each \(j=2,\dots , m\) , \(\epsilon >0\)  and \(\zeta \in 1 _c (\mathcal {O}, \mathbb {R})\)
\(\int _\mathcal {O} \phi _\epsilon \left( \,X_j \zeta +\phi _\epsilon \sum _{s=1}^n b_ {j1 }^{(s)} Y_s \zeta \right) \, d\mathcal {L}^{m+n-1} =- \int _\mathcal {O} D^{\phi _\epsilon }_j \phi _\epsilon \, \zeta \, d \mathcal {L}^{m+n-1}\) 
Using (REF ) and getting to the limit for \(\epsilon \rightarrow 0^+\)  we have that
\(\int _\mathcal {O} \phi \left( \,X_j \zeta +\phi \sum _{s=1}^n b_ {j1 }^{(s)} Y_s \zeta \right) \, d\mathcal {L}^{m+n-1} =- \int _\mathcal {O} D^\phi _j \phi \, \zeta \, d \mathcal {L}^{m+n-1}\) 
Hence \(\phi \)  is distributional solution of \(\left(D^{\phi }_2\phi ,\dots , D^{\phi }_m\phi \right)= w\)  in \(\mathcal {O} \) .
\(\mathbf {(2) \Rightarrow (1)}\) .
Let \(a_0\in \mathcal {O} \) , \(\delta _1>0\)  and \(\mathcal {I}:= \mathcal {U} (a_0,\delta _1) \cap \mathbb {W}\)  and \(\mathcal {I}(a_0, \delta _1)\subset \mathcal {O}\) . Denote
\(K:= \sup _{(x,y)\in \mathcal {I}} \sum _{l=2}^{m}|x_l|,\quad M:=\Vert \phi \Vert _{\mathcal {L}^\infty (\mathcal {I} ,\mathbb {R})}, \quad \delta _2 <\frac{\delta _1}{ 2+\frac{1}{2} K\mathcal {B}_M+M\mathcal {B}_M},\) 
where \(\mathcal {B}_{M} = \max \lbrace  b_{jl}^{(s)} \, | \, l,j=1,\dots , m \, , s=1,\dots , n \rbrace \) .
Peano's Theorem yields that for all \(a=(x_j,\hat{x}_j,y)\in \overline{ \mathcal {I} (a_0,\delta _2)}\)  there exists 1 function \( (\gamma _{j1},\dots , \gamma _{jn}) :[-\delta _2,\delta _2]\rightarrow \mathbb {R}^n\)  such that
\(\gamma ^j_{a}(t)=(x_j+t,\hat{x}_j , \gamma _{j1} (t),\dots , \gamma _{jn}(t))\in \mathcal {I} \quad \mbox{for } t\in [-\delta _2,\delta _2],\) 
and \( \gamma _{js}(t)\)  is a solution of the Cauchy problem
\(\left\lbrace \begin{array}{l}\dot{\gamma }_{js}(t)=\frac{1}{2} \sum _{l=2}^{m}(\hat{x}_j)_l b_{jl}^{(s)}+b_{j1}^{(s)} \phi (\gamma _a^j (t)), \quad \mbox{ for } t\in [-\delta _2,\delta _2]\\\\\gamma _{js}(0)=y_s, \\\end{array}\right.\) 
for \(s=1,\dots ,n\) . It is clear that if \(b_ {j1 }^{(s)}= 0\) , then \( \gamma _{js}(t)=y_s+\frac{1}{2}t \sum _{l=2}^{m}(\hat{x}_j)_l b_{jl}^{(s)}\)  for \(t\in [-\delta _2,\delta _2]\) . On the other hand, if \(b_ {j1 }^{(s)}\ne 0\) , then the map \(\phi (\gamma _a^j (\cdot ))\)  satisfies the following ODE
\(\frac{d}{dt} \left(b_ {j1 }^{(s)} \phi (\gamma _a^j (t)) \right) = b_ {j1 }^{(s)} w_{j} (\gamma _a^j (t)) \\\) 
with \(t\in [-\delta , \delta ]\)  for some \(\delta >0\) . Indeed, we can repeat verbatim the proof of Lemma REF  and so we obtain (REF ) with \(b_{j1}^{(s)}\ne 0\) . Moreover because \(\phi \)  is locally little \(1/2 \) -Hölder continuous along the vertical components, upon dividing (REF ) by \(\epsilon ^2\)  and getting to the limit to \(\epsilon \rightarrow 0\) , we have that
\(\begin{aligned}&\phi (\gamma _a^j (t_2)) - \phi (\gamma _a^j (t_1)) - \int _{t_1}^{t_2} w_j (\gamma _a^j (t)) \, dt = 0 \\\end{aligned}\) 
for \(t_1,t_2 \in [-\delta ,\delta ]\)  with \(\delta >0\)  and \(t_1<t_2\) .
In particular \(\phi (\gamma _a^j (\cdot )) \)  and \(\dot{\gamma }_{js} (\cdot )\)  are \(1([-\delta _3,\delta _3] , \mathbb {R})\)  for \(s=1,\dots , n\)  where \(\delta _3:=\min \lbrace  \delta , \delta _2\rbrace  \) . Therefore the curve \(\gamma ^j_{a}:[-\delta _3,\delta _3] \rightarrow \mathcal {I}\)  satisfies the conditions of the Definition \(\ref {defbroad*}\)  for each \(a\in \mathcal {I} (a_0,\delta _3) := \mathcal {U} (a_0,\delta _3) \cap \mathbb {W}\)  and \(\mathcal {I}(a_0, \delta _3)\subset \mathcal {O}\) .
Then, for each \(j=2,\dots , m\) , \(\exp _{a_0}(\cdot D^\phi _j)(\cdot ):[-\delta _3,\delta _3]\times \mathcal {I} (a_0,\delta _3) \rightarrow \mathcal {I}\)  defined as \(\exp _{a_0}(t D^\phi _{j})(a):= \gamma ^j_{a}(t)\)  is a family of exponential maps at \(a_0\)  which we were looking for. This completes the proof of the implication \((4)\Rightarrow (5)\) .
By Proposition REF  and Theorem \(5.7\)  in [10]}, it immediately follows
Theorem 8.2 
Let \(\mathbb {G}:= (\mathbb {R}^{m+n}, \cdot , \delta _\lambda )\)  be a Carnot group of step 2 and \(, \) W\( the complementary subgroups defined in (\ref {5.2.0}). Let \) : O  be a continuous map, where \(\hat{ \mathcal {O}}\)  is an open subset of \(\mathbb {W}\)  and \(\phi :\mathcal {O} \rightarrow \mathbb {R}\)  is the map associated to \(\hat{\phi }\)  as in \((\ref {phipsi})\) . Then the following conditions are equivalent:

\(\mathrm {graph}\,(\hat{\phi }) \)  is a \(\mathbb {G}\) -regular hypersurface and for all \(a\in \mathrm {graph}\,(\hat{\phi })\)  there is \(r=r(a)>0\)  and \(f\in \mathbb {C}_{\mathbb {G}}^1(\mathcal {U}(a,r)),\mathbb {R})\)  with \(X_1f>0\)  such that
\(\mathrm {graph}\,(\hat{\phi })\cap \mathcal {U}(a,r)=\lbrace  p:f(p)=0\rbrace \) 

\(\hat{\phi }\)  is u.i.d. in \( \hat{\mathcal {O}}\) .

\(D_j^{\phi }\phi \)  interpreted in distributional sense is a continuos function in \(\mathcal {O}\)  and for \(0<\epsilon <1\)  there is a family of functions \(\phi _\epsilon \in \mathbb {C}^1(\mathcal {O}, \mathbb {R})\)  such that for all \(\mathcal {O}^{\prime }\Subset \mathcal {O}\) ,
\(\phi _\epsilon \rightarrow \phi \quad and \quad D_j^{{\phi _\epsilon }} {\phi _\epsilon } \rightarrow D_j^{{\phi }} {\phi }\) 
for \(j=2,\dots , m\) , uniformly on \(\mathcal {O} ^{\prime }\)  as \(\epsilon \rightarrow 0^+\) .

\(\phi \)  is locally little \(1/2 \) -Hölder continuous along the vertical components and there exists \(w\in \mathcal {O} , \mathbb {R}^{m-1})\)  such that \(\phi \)  is a broad* solution of
\(\left(D^{\phi }_2\phi ,\dots , D^{\phi }_m\phi \right)= w, \quad \mbox{ in } \mathcal {O}\) 

\(\phi \)  is locally little \(1/2 \) -Hölder continuous along the vertical components and there exists \(w\in \mathcal {O} , \mathbb {R}^{m-1})\)  such that \(\phi \)  is distributional solution of
\(\left(D^{\phi }_2\phi ,\dots , D^{\phi }_m\phi \right)= w, \quad \mbox{ in } \mathcal {O}\) 


Remark 8.3 We recall that in Theorem \(5.7\)  in [10]}, we consider a locally little \(1/2 \) -Hölder continuous map but in the proof we just use that \(\phi \)  is locally little \(1/2 \) -Hölder continuous along the vertical components.

